Simulation of Highly Efficient Solar Cells

UNIVERSITY OF THE WITWATERSRAND

DOCTORAL THESIS

Simulation of Highly Efﬁcient Solar Cells

Author: Supervisor: Tahir ASLAN Prof. Alexander QUANDT

A thesis submitted in fulﬁllment of the requirement of the degree of Doctor of Philosophy to the Faculty of Science, University of the Witwatersrand, Johannesburg

School of Physics

iii

Declaration of Authorship I hereby declare that this dissertation is my own work. It is being submitted for the Degree of Doctor of Philosophy of Science to the University of the Witwatersrand, Johannesburg. It has not been submitted before to any degree in any other University for assessment purposes.

Tahir ASLAN, October 1, 2018

UNIVERSITY OF THE WITWATERSRAND Abstract Faculty of Science School of Physics

Doctor of Philosophy

Simulation of Highly Efﬁcient Solar Cells

by Tahir ASLAN

In this thesis, we will consider problems causing losses in solar cells and try to solve these problems using numerical methods. We will simulate metal nanoparticles (MNPs) are embedded in order to enhance light absorption for thin ﬁlm solar cells, which otherwise have insufﬁcient absorption of light. To avoid thermal and sub-band losses, we will pick up the idea of using energy conversion processes in single junction solar cells, and discuss modeling of generic up-conversion (UC) and down-conversion (DC) processes, based on rare-earth ions in the context of solar cell device simulations. These numerical simulations are supposed to accompany future experimental studies and practical implementation of such processes in various types of inorganic solar cells. To understand and get parameters that we will need to determine extra current from frequency conversion, the Judd-Ofelt theory [1] has been used.

This work is organised as follows: After describing the basic working principles of a solar cell, we will give an introduction to modern solar cell device simulations, where we discuss the basic equations and simulation parameters [2] and show that most of the key parameters may be taken from ab initio numerical data, rather than experimental data. We will present various numerical approaches, photon absorption/emission processes and a more advanced approach using rate equations [3,4]. Finally we discuss a simple strategy to implement UC and DC layers into solar cell device simulations. Note that this thesis is among the ﬁrst systematic studies of implementing augmenting features into solid state device simulation, apart from crude estimates based on detailed balance models.

vii Acknowledgements

I would like to express my heartfelt gratitude to Professor Alexander Quandt, my supervisor, for giving me permission to undertake this research work and for his guidance on the progress of this work. I would also like to acknowledge Dr. Robert Warmbier for his help to me with computational programs and numerical part of the project. I would also like to acknowledge Prof. J. M. Keartland for their help on administrative issues. I would like to thank Adam Shnier for going through my thesis and discussing the presentation of the science and writing, and for taking over space on my desk to keep his tea cup which he would use often while distracting me to have a good break when I needed it. I would like to thank Itumeleng Mokgosi for always being willing to study together with me. The work discussed in this thesis went smoothly due to the help of my work colleagues. I would like to thank Khalid Mohamed for helping me with ab-initio methods. I would like to thank my wife, Felek Aslan, for her support throughout my time as a PhD student and for bearing with my absence when I attended to my studies. The work discussed in this thesis would not have been possible without the ﬁnancial support from the Materials for Energy Research Group (MERG).

Contents

Declaration of Authorship iii

Abstractv

Acknowledgements vii

1 Introduction1 1.1 Motivation...... 1 1.2 Objective...... 2 1.3 Outline...... 4

2 Fundamental of Solar Cells5 2.1 Introduction...... 5 2.1.1 Properties of Silicon...... 5 Structure...... 5 Band Gap...... 6 P-N junction...... 7 2.2 Current Density...... 8 2.2.1 Dark current...... 8 2.2.2 Photocurrent density...... 9 2.3 J-V Simulation of Ideal c-Si Solar Cell...... 11

3 Device Simulations of Solar Cell 13 3.1 Introduction...... 13 3.2 Simulation Method...... 14 3.2.1 J-V Curve...... 16

4 Plasmonic Nanoparticles 19 4.1 Introduction...... 19 4.2 Dielectric function...... 19 4.2.1 Dielectric function Based on the Drude Model..... 20 4.2.2 Dielectric Function Based on the Drude-Lorentz Model 21 4.3 Scattering and Absorption by Metal Nanoparticles...... 23 4.3.1 Heat generated by plasmonic Nanoparticles...... 30

5 Performance Enhancement of Solar Cells by Plasmonics Nanoparti- cles 35 5.1 Introduction...... 35 5.2 Light Trapping by Metal Nanopatricles...... 36 5.3 Silver and Gold Nanoparticles Embedded Randomly in Silicon 40 x

Optical Properties of Silicon...... 40 5.4 Optical properties of Composite Material...... 41 5.5 J-V Characterisation...... 44

6 Judd-Ofelt Theory 47 6.1 Introduction...... 47 6.1.1 Glasses...... 47 6.1.2 Rare-Earth Ions...... 47 6.1.3 Spectroscopy of Rare-Earth Ions...... 48 Selection rules...... 49 6.2 Analysis of Jodd- Ofelt Theory...... 49

7 Up and Down Conversion and Their Application to Solar Cells 55 7.1 Introduction...... 55 7.1.1 Photon absorption and emission theory...... 55 7.2 Up and Down Conversion...... 57 7.3 Rate Equations for Up/Down-Conversion...... 58 7.3.1 Up-Conversion Case...... 58 7.3.2 Down-Conversion Case...... 61 7.4 Simulation of an Ideal c-Si Solar Cell With and Without UP/- Down Conversion...... 63

8 Conclusions and Future Work 67 8.1 Conclusions...... 67 8.2 Future Work...... 68

A Python Codes for Plotting 69 A.1 Dielectric Function Based on the Drude-Lorentz Model.... 69 A.2 Scattering and Absorption by Metal Nanoparticles...... 70 A.3 Heat Generated by Metal Nanoparticles...... 72

B Publications And Conferences/Workshops Attended 75 B.1 Publications...... 75 B.2 Conferences/workshops Attended...... 76 xi

List of Figures

1.1 Typical losses by an inorganic solar cell. Photons which create thermalisation account for 35 per cent losses. Photons which have less energy than the band gap energy cannot generate the electricity and accounts for a 20 per cent loss (this ﬁgure was obtained from ref. [8])...... 2

2.1 Diamond cubic lattice structure of silicon [25]...... 6 2.2 Band structure of silicon [26]...... 6 2.3 Scheme of absorption and transmission in a solar cell. When a solar cell is illuminated by photons, the electrons can use the energy of photons to jump from the valence band (VB) to the conduction band (CB), creating electron − hole pairs. Electrons with energies larger than the minimum of the conductive band (CB) lose their extra energy through thermalisation. The same happens for holes with energy lower than the maximum of VB.7 2.4 Schema of a p/n-junction with space charge region...... 8 2.5 J-V curve of an ideal c-Si solar cell in the dark, and under illumination. The short circuit current density Jsc is approximately equal to the photo current density Jp and open circuit voltage Voc can be determined from the J − V curve (see Fig. 2. 5). The arrow marks the current density Jm and the voltage Vm at the maximum power Pm...... 12 3.1 Typical one-dimensional setting for solar cell device simulations. 13 3.2 Absorption coefﬁcient of bulk Si, as computed using the Bethe- Salpeter equation and experimental result [34]...... 16 3.3 Simulated J − V characteristics for a silicon p/n-junction solar cell...... 17

4.1 Experimentally imaginary and real part of the dielectric function e for silver, and gold (experimentally data was obtained from ref. [43])...... 20 4.2 Imaginary and Real part of the dielectric function for silver and gold by Drude model...... 21 4.3 Imaginary and Real part of the dielectric function for silver and gold using the Drude-Lorentz model. This was plotted using the python code in the appendix A.1...... 22 4.4 Shifted scattering and absorption cross-section of Ag Np by different media. (r = 30 nm)...... 24 xii

4.5 Scattering and absorption cross-section for Au and Ag Nps as a function of the wavelength, and for various radii of the metallic Nps. The dielectric function of the surroundings is em = 1.8 which is corresponding to ice. This was plotted using the python code in the appendix A.2...... 25 4.6 SEM images of ellipsoidal Ag Nps. The figure was obtained from ref. [52]...... 26 4.7 Schema of prolate and oblate ellipsoid...... 27 4.8 Absorption cross-section of Ag and Au prolate and oblate ellipsoids. (A) In the prolate case; a, b and c are 24 nm, 12 nm and 12 nm, respectively and in the oblate case; a, b and c are 24 nm, 24 nm and 12 nm, respectively and (B) in prolate case; a, b and c are 48 nm, 24 nm and 24 nm, and in oblate case; a, b and c are 48 nm, 48 nm and 24 nm, respectively. em = 2.25...... 28 4.9 Scattering cross-section of Ag and Au prolate and oblate ellipsoids. (A) In the prolate case; a, b and c are 24 nm, 12 nm and 12 nm, respectively and in the oblate case; a, b and c are 24 nm, 24 nm and 12 nm, respectively and (B) in prolate case; a, b and c are 48 nm, 24 nm and 24 nm, and in oblate case; a, b and c are 48 nm, 48 nm and 24 nm, respectively. em = 2.25...... 29 4.10 Temperature changes for different flux intensities as a function of the wavelength in an environment with em = 1.8. The radius r, of the Np is 20 nm...... 31 4.11 (a) Temperature change as a function of distance and (b) temperature change for an array of 3x4 Nps as a function of x, y coor- 8 dinates (RNp = 20 nm). The initial flux intensity is I0 = 1 × 10 W/m2. This was plotted using the python code in the appendix A.3...... 32 4.12 Temperature gradient of one and 12 Nps as a 3x4 array as a function of wavelength at RNp = 70 nm and RNp = 20 nm. The distance between Nps is 100 nm and the initial flux intensity is 8 2 I0 = 1 × 10 W/m ...... 33 5.1 Configuration of random Au Nps in the solar cell (this figure was obtained from ref [57])...... 35 5.2 Extinction cross-section of Au and Ag as a function of radius. 37 5.3 (a) Real and imaginary part of effective dielectric function of Si − Ag composite material and (b) Real and imaginary part of effective dielectric function of Si-Au composite material (Rnp = 10 nm and fs = 0.05)...... 39 5.4 Absorption coefficient of bulk silicon...... 40 5.5 Imaginary and real part of dielectric function and refractive index of bare Si [60]...... 41 5.6 Absorption coefficient of bare bulk silicon, silicon with Ag Nps, and silicon with Au Nps...... 42 xiii

5.7 Absorbance of composite material (a) silicon with spherical gold Nps and (b) silicon with spherical silver Nps (here fs = 0.05, Rnp = 10 nm and L = 1 µm)...... 43 5.8 Absorption coefﬁcient of bare bulk silicon, silicon with Ag Nps and silicon with Au Np (used Eq. 5. 7)...... 44 5.9 J-V curve for thin-ﬁlm c-Si solar cell simulated using GPVDM. 45

6.1 Energy levels of rare-earth ions in the 4 f n conﬁguration. Adopted from ref. [76]...... 50 6.2 Energy level diagram for Dy+3 in a glass with possible transitions. Adopted from ref. [82]...... 52 6.3 Emission cross-section of Dy : CaBA. where the excitation wavelength is λ = 355 nm...... 53 6.4 Energy-level scheme of the Nd+3-doped...... 54

7.1 Absorption and emission process between two energy levels.. 56 7.2 Schematic view of a solar cell with UC and DC layers...... 58 7.3 UC energy level diagram with the most important processes for Tm3+ in the ﬂuoride glass [90]...... 59 7.4 Population density of state N1, N3 and N5 as a function of pump power for the UC system of Eq. 7. 10...... 61 7.5 DC energy level diagram with the important processes for Tb3+-Yb3+...... 62 7.6 J-V for ideal c-Si solar cell, and with UC/DC materials..... 65

List of Tables

2.1 Parameters for the model of a reference current density in Si. 11 2.2 Performance of c-Si solar cell...... 12

3.1 The main input parameters for a thin ﬁlm Si p/n junction solar cell...... 16 3.2 Performance of thin ﬁlm Si solar cell...... 17

4.1 The parameters for a Drude-Lorentz model of Au and Ag ... 22

5.1 The main input parameters for a thin ﬁlm c-Si p/n junction solar cell...... 45 5.2 Performance of a thin ﬁlm c-Si solar cell with and without embedded plasmonic Nps...... 46

6.1 Selection rules for various types of transition in rear-earth ions [74]...... 49 6.2 Necessary parameters for plotting absorption cross-section of Dy : CaBA [84]...... 54

7.1 Parameters for solving the rate equations of Eq. 7. 10...... 60 7.2 Parameters for solving the rate equations of DC [92]...... 63 7.3 Performance of the c-Si solar cell with the UP/DC conversion layer...... 64

xvii

List of Abbreviations

UC Up Conversion DC Down Conversion UV Ultraviolet IR Infrared QE Quantum Efﬁciency VB Valance Band CB Conduction Band DFT Density Functional Theory BSE Bethe Salpeter Equation Np Nanoparticle RE Rare Eearth JO Judd Ofelt GSA Ground State Aabsorption EST Exited State Aabsorption SPA Spontaneous Emission ET Energy Transfer

xix

For Humanity. . .

Chapter 1

Introduction

1.1 Motivation

Solar cells are devices that convert solar photon energy directly into electric energy. Solar energy plays a major role among renewable energy sources, ﬁrst of all it is sustainable and it is environmentally friendly. Among the so-called sustainable energy sources are solar energy, wind power and hydro power [5].

The price of commercial inorganic solar cells has become much cheaper than electricity from coal or nuclear power plants. Furthermore, most energy sources such as, fossil fuels and nuclear have risks which are social, economic, and environmental [6]. For such an important energy technology like solar energy it is important to keep improving performance and reducing costs. This can accelerate the adoption and broaden the use of photovoltaics.

In order to reduce costs and reduce the amount of material used in a solar cell, researchers have used thin film solar cells. The thin film solar cells include second generation solar cells (thicknesses 1 − 2 µm), such as solar cells based on cadmium telluride, copper indium diselenide, amorphous and polycrystalline silicon. Typically these thin film solar cells are more efficient light absorbers, yet because of how thin they are, they do not absorb all of the incident photons. Furthermore not all of the incident photos can be used, and not all of the photons that can be used are actually used very efficiently.

To see how we can improve how single junction solar cells use light we look at the solar spectrum starting with Ultra Violet (UV) and later discussing lower energy photons that are mostly in the infra-red region. Light that comes from the sun spans the UV across the infrared spectral range, but a solar cell such as Si can use photons only from the visible region and the rest is not utilised. The main losses in typical semi-conductor solar cells are due to the thermalisation of electrons in the UV and the visible range of the solar spectrum, the inability to absorb photons with energies below the electronic band gap, and losses due to the recombination of electrons and holes, in particular at the contacts. An ideal solar cell at room temperature may be described as a combination between acting like a black-body absorber and a carnot engine [7] 2 Chapter 1. Introduction

Note that photons which have less energy then the band gap energy are not able to create electron − hole pairs and the corresponding photocurrent, and the photons which have a higher energy than the band gap increase the temperature of the cell. As shown in Fig. 1. 1 out of a 55 per cent loss in energy, 20 per cent is lost due to photons with a sub-band-gap energy (mostly in the infra-red region) and, 35 per cent due to photons with energy above the band gap (for silicon that has a 1.2 eV band gap energy which means that only the photons in visible range may be harnessed with minimal losses). The photons which are useful for generating electricity are the ones that have the energy close to the band gap of the solar cell. Therefore, the ideal single junction solar cell with a single band gap and no resistance losses at the contacts can be at most 35 per cent efﬁcient. But it is not able to use the full photon spectrum [6].

FIGURE 1.1: Typical losses by an inorganic solar cell. Photons which create thermalisation account for 35 per cent losses. Pho- tons which have less energy than the band gap energy cannot generate the electricity and accounts for a 20 per cent loss (this ﬁgure was obtained from ref. [8]).

1.2 Objective

The aim of this study is to understand fundamental losses of solar cells and use numerical methods to suggest how to improve their efficiency towards the 1.2. Objective 3 theoretical limits. The efficiencies of silicon based solar cells are getting closer and closer to the theoretical limit of 34 per cent for single-junction silicon based solar cells, as estimated by Shockley and Queisser in 1961 [9]. These impressive numbers must however be compared to the theoretical efficiency limit of 85 per cent for a generic photovoltaic device [10], which shows that there is plenty of room for improvements.

Frequency conversion processes are not part of standard device simulations, these must be introduced through additional functional layers containing rare-earth ions. After absorption of an incoming photon, the emission of up- or down-converted photons leads to additional photon flux in a frequency range, where the solar cell has much better conversion efficiencies. Thus, the implementation of generic frequency conversion processes in solar cell simulations boils down to the calculation of the additional photon fluxes that result from the frequency conversion processes in these additional functional layers. A promising strategy for single junction solar cells is to convert UV and IR into a frequency range with minimum thermal losses close to the generic electronic band gap (as suggested by Trupke [11]).

The suggested methods are upcoversion (UC) [12] and downconversion (DC) [13]. UC converts lower frequency photons into higher frequency photons. Hence, it converts photons that have sub-band-gap energy, into photons with energies above the band gap. DC converts higher energy photons into less energetic photons which also reduces thermalisation losses (see Fig. 1. 1). This work will investigate UC and DC techniques applied to inorganic solar cells. To get a rough idea about enhancing the efﬁciency of solar cells let us have a look at the photocurrent, Jp, over photon energy, E, is

Z E2 Jp = qQE Φ1.5G(E)dE (1.1) E1

Where QE is the quantum efficiency and Φ1.5G(E) is the solar photon flux on earth. Here, we assume that all photons generate an electron − hole pair. By using this equation we can calculate additional current densities from UC and DC. In the case of UC, the QE is 50 per cent and we integrate the solar spectrum from half of the energy of the band gap which is equivalent to E1 = 0.6 eV to the energy of the full band gap (for Si Eg = 1.1 eV). The A additional photocurrent from the UC is equal to 86 m2 . In the case of DC, QE A is 100 per cent, the additional current is equal to 116 m2 (integration between E1 = 4.4 − E1 = 2.2 eV). We will discuss more comprehensive UC and DC efficiency calculations later on by modeling a conventional Si solar cell and with a layer containing rare earth ions (Tm or Tb and Yb) for the frequency conversion processes.

The other aim is to increase the absorption of light into thin ﬁlm solar cell [14–16], for example embedding metal plasmonic nanoparticles in the cell. 4 Chapter 1. Introduction

The plasmonic nano-particles will increase the absorption of light, and pref- erentially scatter photons into the layer of solar cells. We will study this phenomena in considerable detail. It is also known that the nanoparticles will generate heat affecting their surroundings and we will develop a suite of simulations to predict this phenomena, including the cord ﬁeld effect.

Finally we will also simulate solar cell devices for the development of more efﬁcient types of solar cells. The key parameters will be taken from ab initio numerical data.

1.3 Outline

This thesis discusses numerical simulations to improve the efficiency of solar cells. Chapter 2 presents the fundamentals of solar cells. Chapter 3 presents simulation methods of solar cells. Chapter 4 presents optical properties of nanoparticles. Chapter 5 presents methods of simulation and other techniques that are used for the enhancement of the efficiency of solar cells. Chapter 6 presents the Judd-Ofelt theory and calculates analytically some important parameters such as the life time of photon emissions and furthermore, it gives brief information about the spectroscopy of rare-earth ions. Chapter 7 presents a brief discussion of up-conversion and down-conversion, applied to a c-Si solar cell and calculates how much the efficiency can theoretically be enhanced.

In general, this study should stimulate further experimental and theoretical (numerical) experiments into the use of frequency conversion and plasmonics techniques, with the goal to enhance the efﬁciency of conventional solar cells. 5

Chapter 2

Fundamental of Solar Cells

2.1 Introduction

We will give a brief history of solar cells as a background to the technology. In 1839, the photovoltaic effect was first described by Edmund Bequerel. After 25 years Willoughby Smith described the Effect of Light on Selenium [17]. The first solid state photovoltaic cell was built by Charles Fritts in 1883 but it was only around 1 per cent efficient. After Albert Einstein explained the photoelec- tric effect in 1905 and Vadim Lashkaryov discovered the p-n-junction in 1941 [18, 19], Calvin Souther Fuller, Daryl Chapin and Gerald Pearson invented the first practical photovoltaic cell in 1954. [20, 21]. The scientific community focused only on silicon solar cells until 1990. In the early 1990s multijunction photovoltaic cells used on spacecraft including gallium-arsenide [22].

As mentioned in the previous chapter, a solar cell is a device that converts photon energy into electric energy. The most common commercial type of solar cell is the Si based solar cell. In the following chapter we will develop a reference model for an ideal crystalline silicon (c-Si) solar cell. But before we go through the technical details, we will discuss the fundamental properties of the Si semiconductor which play a signiﬁcant role in a solar cell such as the structure and the band gap of Si. Finally we explain the physics of a typical p/n-junction [23, 24].

2.1.1 Properties of Silicon Structure Silicon (Si) is a group IV element and its crystal structure is a cubic diamond lattice (Fig. 2. 1). Si has the atomic number 14, its atomic weight is 28.085, and the electron orbital conﬁguration is 1s22s22p6[3s23p2]. 6 Chapter 2. Fundamental of Solar Cells

FIGURE 2.1: Diamond cubic lattice structure of silicon [25].

Band Gap Silicon is a semiconductor and it has an indirect band gap, as we can see from Fig 2. 2 [26]. Silicon can conduct heat reasonably well which becomes important to remove thermal energy, which was produced by the thermalisation of charge carriers.

FIGURE 2.2: Band structure of silicon [26].

The band gap of silicon directly affects the efﬁciency of Si based solar cells. Silicon has a band gap of Eg ∼ 1.1 eV and the energy photons coming from the Sun are across a range between ≈ 0.4 to ≈ 6 eV. Fig. 2. 3 demonstrates very well that the photons absorbed have a higher energy than the band gap energy, increase temperature. As the temperature of a solar cell increases due to thermalization of energetic photons the efﬁciency of the solar cell decreases, and the photons that have less energy than the band gap (sub-band gap energy photons) are transmitted through the solar cell without being utilised. Therefore, the ideal case for solar cells is that all photons will have an energy close to the band gap energy. 2.1. Introduction 7

FIGURE 2.3: Scheme of absorption and transmission in a solar cell. When a solar cell is illuminated by photons, the electrons can use the energy of photons to jump from the valence band (VB) to the conduction band (CB), creating electron − hole pairs. Electrons with energies larger than the minimum of the conductive band (CB) lose their extra energy through thermalisation. The same happens for holes with energy lower than the maximum of VB.

P-N junction A p-n junction is formed by contact between p and n-type semiconductors. The p-type has mostly positively charged carriers and the n-type has mostly negatively charged carriers which are induced by dopants that have fewer valence electrons than Si (p-dopants) or more valance electrons than (n-dopants). In practice the doping of silicon with boron (B) results in a p-type and phos- phorus (P) an n-type. When the p-type and n-type are separated or isolated from each other it is a reasonable assumption that, the charge carriers are distributed homogeneously and that the density of the electrons n is equal to the density of ionized donors ND, and density of the holes p is equal to the density of the ionized acceptors NA,

n = ND, p = NA

At thermal equilibrium 2 np = ni (2.1)

Where ni is the so-called intrinsic carrier density. When the p and n types are brought together as in solar cells, the original thermal equilibrium of the free charge carriers in the materials is destroyed through the diffusion of the majority charge across the interface and a new equilibrium is reached, 8 Chapter 2. Fundamental of Solar Cells resulting in an electric ﬁeld E at the junction, called the space charge region. This is shown in Fig. 2. 4.

FIGURE 2.4: Schema of a p/n-junction with space charge region.

2.2 Current Density

In the following we discuss the various methods that describe current flow in a typical inorganic solar cell under an applied electric field. We will introduce important concepts like the dark current Jdark, the photo current Jp and the efficiency of a solar cells.

2.2.1 Dark current From an electronic point of view p/n-junction acts as a diode. At low voltages it lets an electronic current flow primarily in one direction since it has lower resistance in one direction and higher resistance in the other. This leads to a rectification of currents flowing through the device. The size of the space charge region is affected by the magnitude and direction of the electric field. [27]. If an external voltage, V, is applied to a p/n junction in the dark, a dark current density Jdark will be generated, which is given by the diode equation [23, 28] qV k T Jdark = J0(e B − 1) (2.2) 2.2. Current Density 9

where T is temperature, kB is Boltzmann’s constant, q is electron charge and J0 is the saturation current density at V=0 (constant). The latter is given by:

J0 = Jdi f f + Jscr + Jrad (2.3) where Jdi f f is the diffusion current density, which is:

2 Dn Dp Jdi f f = qni ( + ) (2.4) NAln NDlp where ln and lp are diffusion lengths for n and p regions, Dn and Dp are diffusion coefﬁcients for the n and p region, NA is the acceptor doped density, ND is the donor doped density and Jsrc is the recombination current density

wn + wp Jsrc = qni √ (2.5) τnτp where wn and wp the size of the depletion region on the n− and p−side and τn and τp are the lifetimes of electrons and holes in n and p region. Jrad is the radiative recombination current density which takes into account the current density contribution from the direct band gap material [23]. Also note the temperature dependence in Eq. 2. 2.

2.2.2 Photocurrent density The solar cell absorbs photons under illumination which leads to the creation of electron-hole pairs in the space charge region. These electron-hole pairs are separated to be used in an external circuit. Now we describe a p/n- junction under illumination which drives the system out of equilibrium resulting in a photocurrent Jp.

In order to calculate the percentage of photons absorbed by the solar sell we first need to know the reflectivity of and the absorption coefficient of the solar cell. The relation between the reflective index n˜(λ) and the dielectric function e˜(λ) as a function of the photons wavelength λ of a material is given by [29] q n˜(λ) = e˜(λ) (2.6) n˜(λ) has a real and a imaginary part which is related to the real part and the imaginary part of e˜(λ) and can be described by:

n˜(λ) = n(λ) + iκ(λ) (2.7) where the real component n(λ) is the refractive index and the imaginary component κ(λ) is the extinction coefﬁcient, which is proportional to the absorption coefﬁcient σ(λ) [24] by:

κ(λ) σ(λ) = 4π (2.8) λ 10 Chapter 2. Fundamental of Solar Cells

The reflectivity can be calculated from either the dielectric function e˜(λ) or the absorption coefficient σ(λ) as shown in the following text. The fraction of the photons that are absorbed by the solar cell is described by the reflectivity R(λ) of a material. The reflectivity is the fraction of the incident photons that are reflected from the solar cell: 2 n˜ 0 − n˜ 1 R(λ) = (2.9) n˜ 0 + n˜ 1 where n˜ 0 = 1 + i0 is the refractive index of air and n˜ 1 is the complex reflection index of the semiconductor. The transmitted T which is the fraction of the photons that are absorbed by the material is:

T = 1 − R (2.10)

Next we describe the solar photon ﬂux density Φ(E). As we can see from Fig. 1. 1 the sun emits the photons which have an energy including a range from ultraviolet to infrared, and most of the incoming energy is concentrated in the visible range (300 − 800 nm). We can convert the wavelength λ to the energy E of the photon which is given by

E = hc/λ (2.11) where h is Planck’s constant and c is the speed of light. The solar photon ﬂux density Φ(E), is given by [23] ! Z 2F E2 Φ(E) = s dE (2.12) h3c2 E e kBT − 1 where Fs is a geometric factor, which is dependent on the angle between the Sun and surface normal, E is the energy of photons, c is the speed of light. In this case the solar spectrum will be the one on the surface of the earth using the AM 1. 5 (air-mass 1. 5) model, which is the literature standard [30]. The photocurrent is given by [23]

Z E 2 !! 2 2Fs E Jp = q η(E)Ta(E) E dE (2.13) E h3c2 1 e kBT − 1 where η(E) is the probability for the charge carriers to be collected, a(E) is the probability of the absorption of a photon. kB, T and h were deﬁned previously. For the so-called gate maximum, all photons that have energy higher than the band gap energy (E > Eg) are absorbed and each photon in this range generates one electron − hole pair. Which means η = 1 and a = 1. It was assumed that the series and shunt resistance were zero. The basic J − V characteristic of a p-n junction/solar cell may be described by a modiﬁed 2.3. J-V Simulation of Ideal c-Si Solar Cell 11 diode equation including the photocurrent [10]

eV κ T Jnet = Jo e B − Jp (2.14)

Again Jo is a saturation current for V → −∞, and Jp is related to the current generated by the incoming light. Next we will look at a conventional type of Si based solar cell, calculate a model J − V curve and determine the corresponding efﬁciency.

2.3 J-V Simulation of Ideal c-Si Solar Cell

A ideal c-Si solar cell was simulated using Eqs 2. 2, 2. 13 and 2. 14 and the data given in Table 2. 1. Here the dark current density Jdark and photocurrent density, Jp parameters are given by table 2. 1. Fig. 2. 5 shows the Jdark as a function of the applied voltage and the net current Jnet including the photocurrent, and where we were marking the so-called maximum power point.

Parameters Symbol Value −3 16 Intrinsic carrier concentration [m ] ni 15 × 10 −3 23 Acceptor doped [m ] NA 10 −3 25 Donor doped [m ] ND 10 −6 −6 The minor carry diffusion length for n and p region [m] ln and ld 500 × 10 and 10 × 10 2 −3 −3 −4 Diffusion coefﬁcient for n and p region [m s ] Dn and Dp 2.58 × 10 and 2.58 × 1. −7 Charged region of n and p side [m] ωn + ωp 1 × 10 2 −2 −1 −23 Boltzmann’s constant [m kgs K ] κB 1.38 × 10 Temperature [K] T 300

TABLE 2.1: Parameters for the model of a reference current density in Si The photocurrent shown was determined using Eq 2. 13 and was integrated from E1 = 1.1 eV to E1 = 4.1 eV, see Fig. 2. 5. 12 Chapter 2. Fundamental of Solar Cells

FIGURE 2.5: J-V curve of an ideal c-Si solar cell in the dark, and under illumination. The short circuit current density Jsc is approximately equal to the photo current density Jp and open circuit voltage Voc can be determined from the J − V curve (see Fig. 2. 5). The arrow marks the current density Jm and the voltage Vm at the maximum power Pm.

Note that at the maximum working point, JM and VM can be obtained from:

dP = 0 (maximum power) , (2.15) dV Vm Thus using the fact that P = J · V and using Eq. 2. 14 we obtain

κBT 1 + Jp/J0 VM = ln q 1 + qVM/κBT (2.16)

qVm/κBT JM = J0(e − 1) − Jp

The power conversion efficiency of the solar cell is the most important figure of merit to gauge the performance of a solar cell. It is defined by

P η = out × 100% (2.17) Pin

2 where Pout = Jm × Vm is the output power density and Pin = 1000 W/m is input solar power density. From Fig. 2. 5 we get:

Jsc Voc Jm Vm η 440 A/m2 0.67 V ≈ 420 A/m2 ≈ 0.57 V ≈ 24%

TABLE 2.2: Performance of c-Si solar cell. 13

Chapter 3

Device Simulations of Solar Cell

3.1 Introduction

Device simulations are a crucial part of the development of novel and more efﬁcient types of solar cells because they allow us to test an idea or device before the potentially costly and time consuming experimental studies. The necessary simulation parameters are usually taken from experiment. But it turns out that most simulation parameters can be taken from ab-initio data and sometimes experimental data is not needed. In this work we will use data from both experimental and ab-initio sources.

In general the simulation of solar cell devices is mostly considered a one- dimensional problem, where the physical properties are functions of the z-direction perpendicular to the various layers of the device. This is illus- trated in Fig. 3. 1, where light falls on a layered photovoltaic device, which is basically a classical p-n junction with back and front contacts.

FIGURE 3.1: Typical one-dimensional setting for solar cell device simulations. 14 Chapter 3. Device Simulations of Solar Cell

The device is covered by an anti-reﬂective coating on the side exposed to sunlight. Photons may enter the device from this side, and they get absorbed by the active layers of the solar cell. When this happens free charge carriers (electrons and holes) are created. These free charge carriers give rise to the current density J and the voltage V generated by the solar cell. First we discuss the simulation method then the characterisation of the performance of our simulated solar cell.

3.2 Simulation Method

Modern device simulation codes for solar cells predict similar J − V characteristics (and other related physical properties) [2]. To get an idea about the distribution of the electric ﬁelds within a p/n- junction, we have to analyse Poisson’s equation [31]. Poisson equation is

d dϕ −e(z) = e[p(z) − n(z) + N (z) − N (z)] (3.1) dz dz D A

Where n(z) and p(z) are free carrier densities for electrons and holes respectively, e is the dielectric constant, ϕ(z) is the electric potential which gives rise to the overall cell potential V and ND(z) and NA(z) are donor and acceptor densities [32]. The Poisson’s equation can be solved by the iteration method under suitable boundary conditions [31]. The dielectric constant e(z) of the basic materials can either be taken from experiment, or calculated from ﬁrst principles [33], as we will discuss below. In order to determine the corresponding change in electron and hole current density Jn(z), Jp(z), we need to solve two related continuity equations:

1 d (Jn(z)) = −Gopt(z) + R(z); e dz (3.2) 1 d (J (z)) = −G (z) + R(z) e dz n opt

Here Gopt(z), R(z) are the (optical) generation and recombination rates. Note that the generation rates Gopt(z) can be calculated from ﬁrst principles, given the known photon ﬂux Φ(ω, z). This will also be used below. The currents themselves are given by the following expressions:

dn(z) Jn(z) = eµnn(z)E(z) + qDn ; dx (3.3) dn(z) J (z) = eµ p(z)E(z) − qD p p p dx where µn, µp are electron and hole mobilities, which can be taken from first principles calculations [34], Dn and Dp are the diffusion coefficients of electrons and holes respectively and E(z) is electric field. This completes the set of equations to be solved for each point on a path in z-direction, after imposing the proper boundary condition between the different layers [2]. 3.2. Simulation Method 15

The recombination rates R(z) are [2]

4n(z) R(z) = n τ n (3.4) 4p(z) R(z)p = τp

Where 4n(z) and 4p(z) are the excess electron and hole concentration and τn and τp are the lifetime of electrons and holes respectively. The rate at which free charge carries are generated as a function of the energy of photons E, is described by Gopt(E) = α(E)φn(E) (3.5) Where α(E) is the absorption coefﬁcient and φ(E, z) is the solar photon ﬂux (or any other external source of photons). After the substitution of Eq. 3. 4 and 3. 3 into Eq. 3. 2, we get the following second-order differential equations below: d24n(z) 4n(z) D − + G(z) = 0 n d2x τ n (3.6) d24p(z) 4p(z) − + ( ) = Dp 2 G z 0 d x τp

Here the equation can be solved under the appropriate boundary condition to get Jn and Jp [35]. Finally, the total photo-current density is

Jph = Jn + Jp + Jdep (3.7) where Jdep is current density coming from depletion layer (see ref. [31]) The current density is J = Jdark − Jph (3.8) as described in the previous Chapter.

The absorption coefﬁcient α(E) can be determined from ﬁrst principles [33], using Density Functional Theory (DFT). With DFT one can also reliably com- pute all the ground-state properties, which may be derived from the electronic ground state density. But optical properties like α(E) are not based on the ground state. Therefore one needs methods that go beyond DFT, where the optical properties of a material can be calculated either by using the Linear Response Time-Dependent DFT, or based on the Bethe-Salpeter equation (BSE) [36, 37]. Of these two basic approaches, the BSE is the more accurate method, but it is also more cumbersome from a numerical point of view.

An example of such a calculation is shown in Fig. 3. 2. Here we depict the numerical absorption coefﬁcient for bulk Si and a corresponding experimental result which are very close to each other. The accuracy of BSE approach is primarily limited by the computational resources available. Fortunately, 16 Chapter 3. Device Simulations of Solar Cell device simulations are not very sensitive to the ﬁne details of the absorption spectrum, which makes them an ideal application for ab- initio computations.

250 BSE EXP

200

150 m) µ (1/

α 100

0 200 400 600 800 1000 1200 1400 Wavelength (nm)

FIGURE 3.2: Absorption coefﬁcient of bulk Si, as computed using the Bethe-Salpeter equation and experimental result [34].

3.2.1 J-V Curve We now carry out a typical device simulation, using the GPVDM program package [38]. The main input parameters required for such a device simulation are the electric materials and optical properties for each respective layer of the solar cell device (see table 3. 1). Parameters Symbol Value cm2 Electron mobility [ Vs ] µn 1350 cm2 Hole mobility [ Vs ] µp 480 −3 22 Donor doping concentration [m ] ND 10 −3 22 Acceptors doping concentration [m ] NA 10 Static relative permitivity e 11.7 Band gap [eV] Egap 1.15 Temperature [K] T 300

TABLE 3.1: The main input parameters for a thin film Si p/n junction solar cell. Fig. 3. 3 shows the simulation output for a classical silicon p/n junction solar cell. The parameters like the band gap Egap and the static relative permitivity e were taken from the literature [39]. Doping concentration from donors and acceptors were set to NA = ND (moderately doped case). Optical properties were based on absorption coefficients α(E) determined from first 3.2. Simulation Method 17 principles shown in Fig. 3. 1. The simulation J − V characteristics predict an efficiency, η, an open circuit voltage, Voc and a short circuit current density Jsc as given in the table 3. 2.

Jsc Voc η 176 A/m2 0.46 V 6.4 %

TABLE 3.2: Performance of thin ﬁlm Si solar cell.

FIGURE 3.3: Simulated J − V characteristics for a silicon p/n- junction solar cell.

If we compare the simulated devices in Chapter 2 and 3, we see a big difference in efficiency between Table 2. 2 and Table 3. 2. The thin film solar cell has poor absorption because the light has a short path-length in the device and the numerical package that we used also includes a more realistic description of the optical and electronic processes inside the solar cell and prediction of a much lower efficiency.

Chapter 4

Plasmonic Nanoparticles

4.1 Introduction

In a nutshell, plasmonics are the interaction between light and free electrons in a metal. The applications of plasmonics include inter alia light concentration for solar cells, imaging for medicine and cancer treatment [40]. To explore plasmonics, the dielectric function e˜(ω) needs to be determined before we can explain the optical and thermal properties of the metal.

The dielectric function depends on the frequency ω of light and describes the interaction of light with a solid. It can be measured or determined by using numerical modelling. For more details see ref. [41]. Here we restrict, the size of the nanoparticles to between 1 and 100 nm. The size of the nanoparticles should be smaller than the wavelength of light in order to use the dipole approximation. For more extended objects we would be forced to use a coupled-dipole approach [41] which is beyond the scope of this thesis. These small nanoparticles show strong dipolar excitation in the surface plasmon resonances [42]

In this study, we will use both Drude and Drude-Lorentz theory to compare the resulting model dielectric function with the experimental dielectric function. This will be carried out for Ag and Au nanoparticles of different sizes.

4.2 Dielectric function

Note the conversion from angular frequency ω to the wavelength λ is ω = 2πc/λ, where c is the speed of light. The imaginary part of the dielectric function Im(e) describes energy (absorption) losses and the real part of the dielectric function Re(e) describes the phase shift and reﬂection of photons. Fig. 4. 1 shows the experimental result of the real part of the dielectric function Re(e) and the imaginary part of the dielectric function Im(e) of Au and Ag. Re(e) part of Au and Ag describes how light interacts with the metal with Re(e) increasing in magnitude with wavelength, Im(e) part of Au and Ag describes the losses of energy that comes from the electrons motion and which increases with wavelength [41]. From Fig. 4. 1 we see that of the Re(e) for both Au and Ag are negative and they are different due to the difference in 20 Chapter 4. Plasmonic Nanoparticles their electron density (Au has the higher electron density). The Im(e) shows dissipation of energy and differences between them and their resonance frequency.

FIGURE 4.1: Experimentally imaginary and real part of the dielectric function e for silver, and gold (experimentally data was obtained from ref. [43]).

4.2.1 Dielectric function Based on the Drude Model Drude theory is based on a free electron model of a metal. The dielectric function eDrude(ω) is given as:

2 ωp e (ω) = e − (4.1) Drude 0 ω + iΓω where s n e2 ωp = , m e0 is the plasma frequency, and e, m are charge and effective mass of free electrons, e0 is the permitivity of vacuum and n is the electron density. Γ is a damping v f term proportional to l , where v f is the Fermi velocity and l is the electron mean free path between two scattering events [41, 44]. The parameters of 4.2. Dielectric function 21

Au and Ag are given in the table 4. 1. From the parameters in table 4. 1, the dielectric function with its imaginary and real parts is shown in Fig. 4. 2 as a function of wavelength for Au and Ag.

FIGURE 4.2: Imaginary and Real part of the dielectric function for silver and gold by Drude model.

When comparing Figs. 4. 1 and 4. 2, we see that the Drude model gives quite accurate results in the in f rared region, but for UV regions the result is at best an approximation.

4.2.2 Dielectric Function Based on the Drude-Lorentz Model In order to describe the response of the metals to higher energetic photons, we have to include the transition of electrons from the valance band to the conduction band. To get more accurate results we add inter-band contributions to the Drude-Lorentz model, which fi, these take account of bound charges in the metal. With a single one of these interband transitions the dielectric functions is given by [40]

2 2 ωp f ω e(ω) = e − − 1 1 (4.2) ∞ 2 + 2 2 ω iΓpω (ω − ω1) + iΓ1ω 22 Chapter 4. Plasmonic Nanoparticles

where e∞ is the dielectric function at inﬁnite frequency, Γ1 is the Lorentz oscillator damping rate, ω1 is the Lorentz resonance frequency and f1 is a weighting factor. Parameters for Au and Ag are given in table 4. 1.

FIGURE 4.3: Imaginary and Real part of the dielectric function for silver and gold using the Drude-Lorentz model. This was plotted using the python code in the appendix A.1.

Parameters Au Ag ωp [eV] 8.96 9.20 ω1 [eV] 2.97 4.28 Γp [eV] 0.072 0.02 Γ1 [eV] 0.95 0.34 f1 1.78 0.428 e∞ 6.88 3.71

TABLE 4.1: The parameters for a Drude-Lorentz model of Au and Ag

When compared to experimental results, the result from the Drude-Lorentz model (Fig. 4. 3) is a more accurate result, especially in the UV region.

Note that e˜(ω) can also be determined from ab-initio methods based on density functional theory and the data obtained from these simulation can also be used to generate the parameters for the Drude-Lorentz model [45] 4.3. Scattering and Absorption by Metal Nanoparticles 23

4.3 Scattering and Absorption by Metal Nanopar- ticles

Scattering and absorption by plasmonic nanoparticles play a signiﬁcant role for thin ﬁlm solar cells which we will explain in the next chapter. Here we picked Ag with a resonance frequency in the ultraviolet range and Au that resonance frequency in the visible region and calculate the corresponding scattering and absorption cross-sections.

Spherical Case Let us look at spherical nanoparticles Nps with radius, r, embedded in a homogeneous dielectric matrix (this model was ﬁrst examined by Mie[41, 46]) where the radius of the spherical nano-particle will be much smaller than the wavelength of the scattered and absorbed light. Then the absorption and scattering cross-sections are obtain from the polarisability, α as follows [41]

k4 = | |2 σscatt 2 α (4.3) 6πe0 and k σabs = Im[α] (4.4) e0 respectively. The polarisability α of a spherical particle is given by:

3 e(ω) − em α = 4πe0r (4.5) e(ω) + 2em where k is the wavevector in the medium and em is the dielectric constant of the surrounding medium, [14, 47, 48]. We considered only the case of spherical 3D nanoparticles, because lower-dimensional systems do not seem to be of any practical use for plasmon enhanced solar cells.

From Eq. 4. 5 the resonance frequency shifts with the dielectric constant of the medium (see Fig. 4. 4). Here the radius of nanoparticles were ﬁxed at 30 nm. Fig. 4. 4 shows the scattering and absorption cross-section of Ag with different media. 24 Chapter 4. Plasmonic Nanoparticles

FIGURE 4.4: Shifted scattering and absorption cross-section of Ag Np by different media. (r = 30 nm)

Furthermore the cross-section varies with radius, r, of the nanoparticles. 6 3 We see that, σscatt scales with r and σabs scales with r . Therefore, while σabs is dominant for Nps with small radii and, σscatt is dominant for Nps with big radii. This is shown in Fig. 4. 5. We see that absorption is higher than scattering at r = 30 nm as expected from eqs. 4. 3 and 4. 4 for Nps with small radius r. But that should be opposite for Nps with large radius r. For the Ag Np in Fig. 4. 5 at 28 nm the scattering and absorption cross-sections are about equal and for larger particles, the scattering becomes dominant, while for the Au Np at 64 nm they are equal and for larger particles the scattering becomes dominant. We should point out that the radius of the spheres should not be larger than the incident light wavelength, because the electromagnetic ﬁeld that surrounds the Nps should be homogeneous [49–51]. If the Nps are larger we would need to use a different model and the electromagnetic ﬁeld of the light may be focused on a part of the nanoparticle and not be homogeneous. 4.3. Scattering and Absorption by Metal Nanoparticles 25

FIGURE 4.5: Scattering and absorption cross-section for Au and Ag Nps as a function of the wavelength, and for various radii of the metallic Nps. The dielectric function of the surroundings is em = 1.8 which is corresponding to ice. This was plotted using the python code in the appendix A.2. 26 Chapter 4. Plasmonic Nanoparticles

Ellipsoidal Case So far, we have just assumed spherical nanoparticles and we have analyzed their optical properties. In experimental studies the metallic Nps are often not spherical but look like ellipsoids see Fig. 4. 6.

FIGURE 4.6: SEM images of ellipsoidal Ag Nps. The ﬁgure was obtained from ref. [52].

In this section we want to study small and ellipsoidal metal nanoparticles and compare the results to the spherical case. The polarisability of ellipsoidal Nps is given by e(ω) − e α = V m (4.6) 1−Li e(ω) + em Li 4π where V = 3 abc (a,b and c are the radii of axes) is the volume of the ellipsoidal nanoparticle, and Li is a geometrical factor with i = x, y or z. Two special ellipsoidal shapes are the prolate and the oblate ellipsoid. As can be seen from Fig. 4. 7 for the oblate ellipsoid the relationship between coordinates is x = y > z and for the prolate ellipsoid is z > x = y. 4.3. Scattering and Absorption by Metal Nanoparticles 27

FIGURE 4.7: Schema of prolate and oblate ellipsoid.

Fig. 4. 7 shows simulation result the prolate and the oblate ellipsoids that will be considered as possible metal Nps in our study. The geometric factor along the x direction, Lx, for the prolate ellipsoidal and oblate ellipsoidal nanoparticles are given as, [53],

1 − e2 1 1 + e L = − 1 + ln x,prolate e2 2e 1 − e and (4.7) 1 − e2 1 1 + e L = − 1 + ln x,oblate e2 2e 1 − e q = − c2 Where e 1 a2 . The other geometric factors Ly and Lz in the y and z direction are related to Lx, as follows

L = L = (1 − L )/2 > 1/3 > L for prolate y z x x (4.8) Lz = 1 − 2Lx > 1/3 > Lx = Ly for oblate

We can now use Eqs. 4. 3, 4. 4 and 4. 6, to determine the absorption cross- section for both prolate and oblate ellipsoids. Fig 4. 8 shows the absorption cross-sections for Ag and Au prolate and oblate nano-ellipsoids perpendicular to different axes. 28 Chapter 4. Plasmonic Nanoparticles

(A)

(B)

FIGURE 4.8: Absorption cross-section of Ag and Au prolate and oblate ellipsoids. (A) In the prolate case; a, b and c are 24 nm, 12 nm and 12 nm, respectively and in the oblate case; a, b and c are 24 nm, 24 nm and 12 nm, respectively and (B) in prolate case; a, b and c are 48 nm, 24 nm and 24 nm, and in oblate case; a, b and c are 48 nm, 48 nm and 24 nm, respectively. em = 2.25.

From Eqs. 4. 4 and 4. 6, we have plotted the absorption cross-section for both the prolate and the oblate ellipsoid. Fig 4. 9 shows the scattering cross-section for Ag and Au prolate and oblate nano-ellipsoids in different directions. 4.3. Scattering and Absorption by Metal Nanoparticles 29

(A)

(B)

FIGURE 4.9: Scattering cross-section of Ag and Au prolate and oblate ellipsoids. (A) In the prolate case; a, b and c are 24 nm, 12 nm and 12 nm, respectively and in the oblate case; a, b and c are 24 nm, 24 nm and 12 nm, respectively and (B) in prolate case; a, b and c are 48 nm, 24 nm and 24 nm, and in oblate case; a, b and c are 48 nm, 48 nm and 24 nm, respectively. em = 2.25.

Compared to the spherical case, the cross-sections for the ellipsoidal case vary with the size of Nps and show the same behaviour as a function of increasing size for absorption and scattering cross-section see Fig 4. 8 and 4. 9. 30 Chapter 4. Plasmonic Nanoparticles

4.3.1 Heat generated by plasmonic Nanoparticles When plasmonic Nps are embedded in a medium that is illuminated by light, they will generate heat when the photons are absorbed. We just used the simplest possible model to describe heating.This property of plasmonic Nps will decrease the solar cell’s efﬁciency but it is beneﬁcial for other applications such as medicine [54]. For this section, we will only use the Au Nps but the same simulation can be done for other metals as well.

If we assume that there is no the phase transformation, the temperature distribution around the Nps is determined by the general heat transfer equation: dT(r, t) p(r)c(r) = ∇k(r)∇T(r, t) + Q(r, t), (4.9) dt where T(r, t) is the local temperature, which depends on the coordinate, r, and time t. p(r), c(r), k(r) and Q(r, t) are the mass density, speciﬁc heat, thermal conductivity, and an energy source Q term that stems from light dissipation in the Nps. The energy source Q from the dissipation of light in the Nps is related to the absorption cross section, and for spherical particles we obtain [50] ω 2 3em 2 Q = E0 | | Im(eNP), (4.10) 8π 2em + eNP where E0 is the amplitude of the incoming radiation, em is the dielectric constant of the surround medium, and eNP is the dielectric function of the Np. The corresponding intensity is given by

2√ I(t) = I0 = cE0 em/8π (4.11)

The analytical result of Eq. 4. 9 under steady-state conditions is easily obtained for the local temperature proﬁle on and surrounding a single Np is

( VNPQ for r = RNp ∆T(r) = 4πk0RNp (4.12) VNPQ for r > R 4πk0r Np where ∆T is the temperature difference relative to the initial state without illumination, r is the distance from the center of the spherical Np, k0 is the thermal conductivity of the surrounding medium, and VNP is the Np volume. From the Eqs. 4. 10, 4. 11 and 4. 12 we see that the temperature difference is proportional to the photons ﬂux intensity (see Fig. 4. 10). Fig. 4. 10 shows the temperature difference as a function of the intensity. 4.3. Scattering and Absorption by Metal Nanoparticles 31

FIGURE 4.10: Temperature changes for different ﬂux intensities as a function of the wavelength in an environment with em = 1.8. The radius r, of the Np is 20 nm.

The fig. 4. 10 shows us the plasmon resonance wavelength, which is clear in the figure as λ = 520 nm. The temperature change on the Np and its surroundings for one 20 nm Au Np can then be determined from the equations given above. The results are shown in Fig. 4. 11 (a) where the temperature change on and around the Au Np is plotted as a function of distance from the nanoparticle. In fig. 4. 11 (b) is a map of the temperature change in a 2 dimensional plane sliced from a 3 dimensional model where 12 Nps (3x4) sit on this plane. The distance between the Nps is 80 nm. 32 Chapter 4. Plasmonic Nanoparticles

(A)

(B)

FIGURE 4.11: (a) Temperature change as a function of distance and (b) temperature change for an array of 3x4 Nps as a function of x, y coordinates (RNp = 20 nm). The initial ﬂux intensity is 8 2 I0 = 1 × 10 W/m . This was plotted using the python code in the appendix A.3. 4.3. Scattering and Absorption by Metal Nanoparticles 33

From Eq. 4. 12 we can see the temperature changes from the surface of the Np into the bulk with radius of the Np,RNp, as follows:

2 ∆T ∼ RNp (4.13)

Now we will look at the temperature drops from the surface of the nanoparticle in to the bulk. Fig. 4. 12 shows the temperature gradient for one and many Nps with two different radii. This is to illustrate Eq. 4. 13 above.

FIGURE 4.12: Temperature gradient of one and 12 Nps as a 3x4 array as a function of wavelength at RNp = 70 nm and RNp = 20 nm. The distance between Nps is 100 nm and the initial ﬂux 8 2 intensity is I0 = 1 × 10 W/m .

Note that we only considered the heating caused by higher concentrations of plasmonic nanoparticles other heating effects are secondary in this respect.

Chapter 5

Performance Enhancement of Solar Cells by Plasmonics Nanoparticles

5.1 Introduction

In this section we will simulate the increase in the efficiency of a solar cell using embedded plasmonic nanoparticles. We will be considering thin film solar cells, they are relatively cheaper than the conventional solar cells and they are easier to fabricate. Furthermore because they are thinner there is less charge carrier recombination than in a conventional solar cell. Their main disadvantage however is that there is insufficient absorption of light thus reducing their efficiency.

Fig. 5. 1 shows the structure of random Nps in such a thin ﬁlm solar cell, (see ref. [55, 56]).

FIGURE 5.1: Conﬁguration of random Au Nps in the solar cell (this ﬁgure was obtained from ref [57]). Chapter 5. Performance Enhancement of Solar Cells by Plasmonics 36 Nanoparticles

The numerical and experimental method that we will use is the direct embedding of metal Nps, such as Au or Ag, into the solar cell to enhance light collection. Once embedded the Nps will cause extra light absorption. This leads to the creation of additional electron − hole pairs within the active layer of solar cells and therefore to an extra current density.

The key will be the determination of the effective absorption coefﬁcients of bare Si solar cells due to the deposition of metal Nps. On that basis we will show how to estimate the efﬁciency of these augmented solar cells.

5.2 Light Trapping by Metal Nanopatricles

We have already calculated the optical properties of metal Nps in previous chapters, such as the dielectric function, absorption cross-section and scattering cross-section. The concentration, shape and size of Nps plays a signiﬁcant role in these processes, which was partly demonstrated in the previous chapters.

In the following Fig. 5. 2 we will plot the extinction cross-section as a function of the nanoparticle radius and the dielectric function of spherical Ag and Au Nps in a frequency range of interest for photovoltaics. Extinction cross-sections σext are given by

σext = σabs + σsct (5.1) where σabs and σsct are absorption and scattering cross-sections respectively. Fig. 5. 2 shows the extinction cross-section of Au and Ag as a function of radius. Evidently, the Ag surface resonance is at 350 nm which is in the UV range, and the values are higher than for Au, where the surface resonance is at 500 nm in the visible range. 5.2. Light Trapping by Metal Nanopatricles 37

FIGURE 5.2: Extinction cross-section of Au and Ag as a function of radius.

Note that the dielectric function, e of spherical Au and Ag is given by

2 2 ωp f ω e(ω) = e − − 1 1 (5.2) ∞ 2 + 2 2 ω iΓpω (ω − ω1) + iΓ1ω based on the Drude-Lorentz theory. This has been derived in the previous chapter, where values for the different parameters are indicated.

Now let us assume that the Nps are embedded inside a thin ﬁlm Si solar cell forming a composite. Some of the best ways to get information about the optical properties of a composite material are ﬁrst principle methods or experimental results; however we will use effective medium theory in this chapter. Effective medium theory as developed by Maxwell-Garnett to calculate the effective dielectric function ee f f of a composite [57, 58].

We assume that N spherical Nps are distributed randomly across the silicon solar cell. Then the effective dielectric function

 ( − ) 2 2 fs es e es−e e(1 − f )4 1 + ( + ) = es 2e + 3 3 es+2e ee f f 2  ( − ) i2 fsk a ( − )  (5.3) (1 + 2 f ) 1 − fs es e 1 − fs es e (es+2e) es+2e where e is the dielectric function of the medium, es is the dielectric function of Chapter 5. Performance Enhancement of Solar Cells by Plasmonics 38 Nanoparticles

the metal Nps, fs is the the ﬁlling fraction of the metal in the composite which is proportional to the number of the metal Nps N, k is the wavenumber, and a is the radius of the Nps. Fig. 5. 3 shows the effective dielectric function for a typical Si − Ag and Si − Au composite material. It is quite obvious that Ag and Au Nps have different optical behaviors when embedded into silicon as we can see from Fig. 5. 3 [57]. 5.2. Light Trapping by Metal Nanopatricles 39

(A)

(B)

FIGURE 5.3: (a) Real and imaginary part of effective dielectric function of Si − Ag composite material and (b) Real and imaginary part of effective dielectric function of Si-Au composite material (Rnp = 10 nm and fs = 0.05). Chapter 5. Performance Enhancement of Solar Cells by Plasmonics 40 Nanoparticles

5.3 Silver and Gold Nanoparticles Embedded Ran- domly in Silicon

Now we will study the enhancement of the efficiency of thin film Si solar cells. In order to calculate the corresponding current density we first have to determine the absorption coefficient of the composite material.

Optical Properties of Silicon The absorption coefﬁcient and the dielectric function of bare bulk Si may be obtained as follows: The absorption coefﬁcient of bulk Si is calculated from the k-selection rule here, given by [59]

α(ω) = 0.0287exp[2.72(h¯ ω − Eg)] (5.4) where h¯ ω is the energy of an incident photon which is > 1.5 eV for this condition, and Eg is the band gap of silicon. We can plot this absorption coefﬁcient of bare silicon (Eq. 5. 4) and we can compare it to silicon with Ag or Au Nps embedded, the results follow,

FIGURE 5.4: Absorption coefﬁcient of bulk silicon.

We have also absorption coefﬁcient data from ab-initio for a bulk silicon in Chapter 3. Fig 3. 2. It is also useful to know the complex dielectric function of the bare bulk Si as well, which is shown in the Fig. 5. 5. 5.4. Optical properties of Composite Material 41

FIGURE 5.5: Imaginary and real part of dielectric function and refractive index of bare Si [60].

5.4 Optical properties of Composite Material

The absorption coefﬁcient of a composite material such as Si − Ag and Si − Au may be calculated from the dielectric function or the complex refractive index. The relationship between the refractive index and the dielectric function was determined in the previous chapter. Using this relationship, we can determine the absorption coefﬁcient by taking the square root of the dielectric function and then determining α from

4πκ α(m−1) = λ or (5.5) −1 α(m ) = 4πIm(ee f f )/λ where is λ is mean free path [29]. The absorption coefﬁcient with and without Nps is shown in Fig. 5. 6. Chapter 5. Performance Enhancement of Solar Cells by Plasmonics 42 Nanoparticles

FIGURE 5.6: Absorption coefﬁcient of bare bulk silicon, silicon with Ag Nps, and silicon with Au Nps.

Fig. 5. 6 we can see the enhancement of optical absorption for both Si − Ag and Si − Au in the visible region.

Now we will look at the absorbance A of this composite structure. This is related to the amount of photons absorbed by the structure, which are extracted as current by an external circuit attached to the solar cell.

A also depends on the effective medium, it is given by [61]

A = 1 − exp(−α(λ)L) (5.6)

Here L is the thickness of the absorbing layer. Fig. 5. 7 shows the absorbance for Si − Ag and Si − Au Nps composite materials. 5.4. Optical properties of Composite Material 43

FIGURE 5.7: Absorbance of composite material (a) silicon with spherical gold Nps and (b) silicon with spherical silver Nps (here fs = 0.05, Rnp = 10 nm and L = 1 µm).

The absorption coefﬁcient can be determined in the case of spherical and small Nps, using the approximate expression [62] r 8.88 q α(m−1) = −Ree + Ree2 + Ime2 (5.7) λ e f f e f f e f f Chapter 5. Performance Enhancement of Solar Cells by Plasmonics 44 Nanoparticles

FIGURE 5.8: Absorption coefﬁcient of bare bulk silicon, silicon with Ag Nps and silicon with Au Np (used Eq. 5. 7).

We plotted the resulting absorption coefﬁcient from Eq. 5. 7 in Fig. 5. 8. The result looks slightly different compared to Fig. 5. 6 from Eq. 5. 5 [63]. The Fig. 5. 8 shows the absorption coefﬁcient of bare bulk silicon, silicon with Ag Nps and silicon with Au Nps.

So far we have used only spherical metal Nps, but the same methods can be applied for non-spherical metal Nps as well as for ellipsoids [64].

5.5 J-V Characterisation

After determining the absorption coefﬁcient, we can determine the generation for electron − hole pairs (using Beer’s low) given by [24]

G(λ, x) = φ(λ)α(λ)e−α(λ)x (5.8) where x is the length of the absorbing material and α(λ) is the absorption coefficient of the composite material. Here φ(λ) is the solar photon flux density. Furthermore we assume that all photons pass into the solar cell (no reflection). The enhancement of the photocurrent current density from the 5.5. J-V Characterisation 45 composite solar cell can be calculated using

Z λ2 Z L ∆Jp = −q G(λ, x)dx (5.9) λ1 0 where q is the elementary charge. We integrated over wavelength from λ1 = 300 nm to λ2 = 1100 nm. Our calculations showed better performance for silicon with silver spherical Nps as compared to silicon with gold spherical Nps. Fig. 5. 9 shows the current voltage curve of the bare silicon solar cell, and for spherical Ag and Au Nps embedded into a silicon cell. In order to predict the J − V curve the General-purpose Photovoltaic Device Model (GPVDM) simulation program was used [38]. The resulting J − V curve for a thin ﬁlm c-Si solar cell is given by Fig. 5. 9 and as can calculate from the J − V curve Jsc,Si = 226.26 open circuit voltage is Voc = 0.454 and all necessary parameters were given in table 5. 1.

Parameters Symbol Value m2 7 Electron mobility [ Vs ] µn 5 × 10 m2 6 Hole mobility [ Vs ] µp 4.5 × 10 −3 26 Donor doping concentration [m ] ND 5 × 10 −3 26 Acceptors doping concentration [m ] NA 5 × 10 Static relative permittivity e 11.7 Band gap [eV] Egap 1.6 Temperature [K] T 300

TABLE 5.1: The main input parameters for a thin ﬁlm c-Si p/n junction solar cell.

FIGURE 5.9: J-V curve for thin-ﬁlm c-Si solar cell simulated using GPVDM.

To analyse the performance of a thin-ﬁlm c-silicon solar cell, the data in table 5. 1 has been used to calculate the efﬁciency η. The enhancement of short Chapter 5. Performance Enhancement of Solar Cells by Plasmonics 46 Nanoparticles

−2 circuit current density for the Si − Au composition ∆Jsc,Si−Au is 104 Am and −2 for Si − Ag composition ∆Jsc,Si−Ag is 44 Am To calculate the efﬁciency, η we will use same method that we used in the previous chapter. It was deﬁned as

P η = out × 100% (5.10) Pin

2 where Pin = 1000W/m is input solar power density and output power density Pout = Jm × Vm.

−2 Jm,Si 219 Am −2 Jm,Si−Au ∼ 290 Am −2 Jm,Si−Ag ∼ 350 Am Vm,Si 0.37 V Vm,Si−Au ∼ 0.37 V Vm,Si−Ag ∼ 0.37 V ηSi 8.1 % ηSi−Au 13.6 % ηSi−Ag 11 %

TABLE 5.2: Performance of a thin ﬁlm c-Si solar cell with and without embedded plasmonic Nps.

In order to calculate the efﬁciency enhancement from the composite materials, we added up the additional current density due to the plasmonic nanoparticles to the Jsc of solar cell without plasmonic nanoparticles to approximate Vm,Si−Au, Vm,Si−Ag, Jm,Si−Au and Jm,Si−Ag. The calculation results given by table 5. 1. We predict 2.9 per cent enhancement from Si − Ag and 5.5 % for Si − Au. This is based on calculations using 10 nm Nps which are suitable to show plasmonic effects, the Np size can be optimized in future work. 47

Chapter 6

Judd-Ofelt Theory

6.1 Introduction

As we discussed in the previous chapter, the losses in efﬁciency of a single- junction solar cell was over 40 per cent because of thermalisation and transmission [65]. Here we will discuss methods to reduce these losses, using frequency conversion techniques based on rare earth ions. To this end a solar cell will be augmented by extra layers of glass which are doped with rare-earth (RE) ions. These extra layers are placed on the top and back of solar cells. In the following section we will give some background information about glass layers containing RE ions.

6.1.1 Glasses Due to the high transparency of glass for visible light it is very commonly used in daily life and technology like optical instruments, windows and photonic devices [65, 66]. Moreover, its fabrication is easy and cheap. The sol-gel technology can be used for the fabrication of quality glass and it is often the experimental method of choice to implement frequency conversion techniques in solar cells [67]. Note that the glass for frequency conversion applications requires low optical scattering and low ionic/atomic absorption. For more information see ref. [65]

6.1.2 Rare-Earth Ions In this study, RE ions are used in a composite with glass to enhance the efﬁciency of solar cells. The RE ions let the cells make better use of light which is outside the visible range. An amazing property of rare-earth (RE) ions is their ability to convert light from near-infrared or UV range into the visible range. Therefore, there are many practical applications in optics based on RE ions [68–70]. Below, we give a short introduction to Judd-Ofelt (JO) theory, which describes characteristics of the transition properties of RE-glass, including important concepts like oscillator strengths, branching ratio, lifetime, intensity parameters and transition probability [71]. 48 Chapter 6. Judd-Ofelt Theory

6.1.3 Spectroscopy of Rare-Earth Ions Spectroscopy studies are based on the interaction between light and matter, and it is particularly concerned with how matter will absorb or emit light of different frequencies. The energy of this absorbed or emitted light is related to electrical and structural characteristics of matter.

This way one can also get information about single atoms such as, the valence or excited state of electrons in an atom. It is useful to know how to identify the energy levels of an atom. Atomic states are usually labeled by the symbol ,

2S+1 LJ, where S is the total spin quantum number, L is the label of the total orbital angular momentum and J is the total angular momentum (l + s ≥ j ≥ l − s) The ground state of an element like Erbium (Er) is inﬂuenced by the spin-orbit coupling, the Pauli exclusion principle and Hund’s rule. These rules tell us how the electrons are distributed in the available orbitals (for more detail about the Hund rules and Pauli exclusion principle see ref. [72]). The electronic conﬁguration of an Er atom for example is [Xe] 6s2:4 f 12. Let us apply three Hund rules to determine the corresponding spectroscopic symbol. Here are three Hund rules, respectively

1. Rule To ﬁlling the shells

Er : [Xe]6s2 : 4 f 12 = ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑ ↑ ml = 3 2 1 0 -1 -2 -3 2. Rule To sum over spin up and down

7 S = ∑ Si = 1 i=1 Total spin is 2S + 1 = 3 which is triplet level

3. Rule To sum over both spins

7 l = ∑(ml)i = 5 i=1 which is the H-term in spectroscopic notation. The subshell is more than half ﬁlled so the maximum value of J is, s + l = 5 + 1 = 6. This is used to specify 2S+1 3 the ground state electron conﬁguration, then LJ = H6 [73].

Following the same method the ground state electron conﬁguration of Er3+ 4 ion can be stated as I15/2. 6.2. Analysis of Jodd- Ofelt Theory 49

Selection rules Optical transitions in RE ions are characterised by the selection rules tabulated in table 6.1 for a free ion and in in the presence of an external electric ﬁeld. These rules will change for high applied electric and magnetic ﬁelds. [71].

Free ion case External ﬁeld case

Angular momentum L: ∆ L Forced electric dipole ∆l= = 0, ± 1 ±1, ∆L, ∆J ≤ 2l, ∆S = 0 for 0 ↔ 0 transition

Total angular momentum J: Forced magnetic dipole ∆L ∆J = 0, ± 1 = 0, ∆J = 0, ± 1, ∆S= 0

Spin quantum number S: Quadrupole ∆J= 0, ≤ ±2 ∆S= 0

Magnetic dipole transition M: ∆L = 0, ± 1, ± 2, ∆J = 0, and ∆S = 0

TABLE 6.1: Selection rules for various types of transition in rear-earth ions [74]. The quantum numbers L, J, S and M describe the state of electrons a material.

6.2 Analysis of Jodd- Ofelt Theory

Jodd-Ofelt (JO) theory analysis describes the intensities of f-f transitions in the spectra of lanthanide compounds [71].

Before we go into the details let us describe some of the properties of RE. There are two types of rare-earth ions; the first type is divalent with electronic, 4 f N5s25p6, configuration, and the second type is trivalent with 4 f N−15s25p6 electronic configuration. Some typical divalent rare earth ions are Europium (Eu2+), Ytterbium (Yb2+), Samarium (Sm2+) and some trivalent rare earth ions are Thulium (Tm3+), Erbium (Er3+), Praseodymium (Pr3+), Cerium (Ce3+). The electrons in the f shell are shielded by the 5s, p, d shell which leads to atomic-like spectroscopy properties of these inner shells . The Fig 6. 1 shows the energy level diagram of RE ions. Because of a weak crystal field effect the energy levels of free ions and in crystal are similar. For a discussion of the corresponding Hamiltonian see [75]. 50 Chapter 6. Judd-Ofelt Theory

FIGURE 6.1: Energy levels of rare-earth ions in the 4 f n conﬁgu- ration. Adopted from ref. [76]

Experimental oscillator strength values (for f-f transitions) for crystals are usually found by integrating the absorption band of a RE doped crystal as [77], Z −9 −1 fexp = 4, 32x10 (cd) σ(λ)dσ, (6.1) where c is the concentration of the lanthanide ions in mole−1 and d is the optical path length. The oscillator strengths may also be calculated using

8π2mcν (n2 + 2)2 f = Ω |h f nψJ||U(λ)|| f nψ0 J0i|2, (6.2) cal ( + ) ∑ λ 3h 2j 1 9n λ=2,4,6 6.2. Analysis of Jodd- Ofelt Theory 51 where m is the mass of electron, c is the speed of light in vacuum, h is Plank’s constant, n is the refractive index of the surrounding medium, J is the total angular momentum of the initial state ν is the transition energy and Ωλ is the JO intensity parameters. Transitions between J multiplets are valid for only three JO intensity parameters Ω2, Ω4 and Ω6. The value of these parameters are extracted from experimental results by ﬁtting theoretical oscillator strengths to the experimental data[78]. These Ωλ parameters help to analyse the relations between spectra and structural properties of lanthanide complexes of different kinds. Note that the matrix elements were tabulated by Carnall et al. [79] and JO intensity parameter Ω2, Ω4 and Ω6, are determined by a least squares ﬁt using experimental data[80, 81].

The oscillator strength is calculated based on Eq. 6. 2, and it is proportional to the electric dipole strength Sed

n (λ) n 0 0 2 Sed = ∑ Ωλ|h f ψJ||U || f ψ J i| (6.3) λ=2,4,6 and the magnetic dipole Smd strength

h2 S = |h f nψJ|L + 2S| f nψ0 J0i|2 (6.4) md 16π2m2c2 The radiative transition probability is the total matrix element of the electric dipole plus the magnetic dipole transitions (ED + MD) between an initial J and ﬁnal J state. It is given by,

64π4e2 n2 + 2 ( 0 ) = ( )2 + 2 A J , J 3 n SED n SMD (6.5) 3h(2J0 + 1)λ 3

Where e is the electron charge (in electrostatic unit) and λ is the mean wavelength. The radiative lifetime τr, and the branching ratio, β [71], can be calculate by the following equations;

1 = ∑ A(J0; J) (6.6) τr J

A(J0; J) βJ0;J = (6.7) τr These parameters are important when calculating the lifetime of transition processes and the intensity of emission lines that are used for UC and DC conversion as discussed in the following chapter. The emission cross-section σ(λ) is λ4 ( ) = ( 0 ) σ0,abs λ 2 A J ; J (6.8) 8πcn ∆λe f f 52 Chapter 6. Judd-Ofelt Theory

Where ∆λe f f is a line shape function which may be obtained from

R I(ν)dν ∆λ = . (6.9) e f f I Note that I(ν) is irradiance as a function of frequency ν. A frequency dependent cross-section may be obtained from

Γ2/4 ( ) = σ ω abs σ0,abs 2 2 (6.10) (ω − ω0) + Γ /4

Where Γ = A(J0; J) and ω = 2πc/λ.

After going through JO theory we can determine the absorption coef- ﬁcient of RE ions. Here rare earth ion is Dy3+ and embedded in transparent materials such as glass-CaBA. CaBA that plays a role as a host material. This material is illuminated with 355 nm wavelength by a Nd:YAG laser. The material absorbs the laser light (λ = 355 nm) and re-emits three different wavelengths that correspond to 660 nm (4 f9/2 → 6H11/2), 575nm (4 f9/2 → 6H13/2) and 475nm (4 f9/2 → 6H15/2)[82] (see Fig. 6. 3). This is a

+ FIGURE 6.2: Energy level diagram for Dy 3 in a glass with possible transitions. Adopted from ref. [82] signiﬁcant process that converts high energy photons in the UV region to up to three lower energy photons in the visible region. This process is photon downconversion. Such a layer would reduce the thermalisation of electrons which normally increases the temperature of the device and for this reason the efﬁciency will increase.

Fig. 6. 3 shows the corresponding absorption cross-sections (by Eq. 6. 10), using the parameters given in table. 6. 2. They correspond to 6.2. Analysis of Jodd- Ofelt Theory 53

4F15/2 → 6H11/2 (at 673nm wavelength), 4F15/2 → 6H13/2 (at 585nm wavelength) and 4F15/2 → 6H15/2 (at 494nm wavelength)

FIGURE 6.3: Emission cross-section of Dy : CaBA. where the excitation wavelength is λ = 355 nm.

From the relationship between the emission and absorption cross-section that was calculated using the McCumber theory [83], we can determine the emission cross-section σemis(ω). The relationship is given by e − hν σemis(ω) = σabs(ω)exp . (6.11) kbT Where e is the energy for a transition from the ground state to an excite state, ν is the photon frequency of the emitted photon, T is the temperature, h is the Planck constant and kb is the Boltzmann’s constant. 54 Chapter 6. Judd-Ofelt Theory

4F9/2 → 6H11/2 4F9/2 → 6H13/2 4F9/2 → 6H15/2

−26 2 −26 2 −26 2 σ0,abs 5.49 × 10 m ) 37.2 × 10 m ) 5.81 × 10 m )

A(J’;J) 0.089 0.853 0.264

λ0 660 nm 575 nm 475 nm

n 1.65

TABLE 6.2: Necessary parameters for plotting absorption cross- section of Dy : CaBA [84].

By a similar technique , we can convert the light from the infrared region to the visible region by a process of up-conversion (UC) see Fig 6. 4. Fig 6. 4 shows the schema of UC. Here using an excitation wavelength 780 nm the following emission wavelengths are observed: 374 nm, 547 nm, 656 nm and 714 nm.

+ FIGURE 6.4: Energy-level scheme of the Nd 3-doped

In the following chapters, we will discuss two different methods for increasing the efﬁciency of the solar using UC and DC. 55

Chapter 7

Up and Down Conversion and Their Application to Solar Cells

7.1 Introduction

In the previous chapter, we discussed plasmonic nanoparticles as a new approach to increase the efﬁciency of solar cells. Now we will use another approach based on up-conversion (UC) and down-conversion (DC). With UC/DC we basically modify the solar spectrum experienced by the solar cell. Standard conventional solar cells cannot utilise all photons in the solar spectrum. Photons with sub-band-gap energies cannot be absorbed by conventional Si solar cells, and 20 per cent of the solar energy that reaches earth’s surface is wasted [16, 85]. Other losses are from photons with excess energy (w.r.t the band gap). They lose their excess energy through thermalisation, causing heating of the solar cell which decreases their efﬁciencies [86].

In order to prevent losses from photons, with energies largely different from the electronic band gap, we will use UC and DC methods respectively. To supply a solar cell with UC/DC one of the easiest techniques is to implement layers made of a luminescent material such as a glass layer doped withe rare-earth ions which is placed on top (DC) or at the bottom (UC) of a solar cell as a extra layer.

Theoretically, the efﬁciency of silicon solar cells with DC can be in- creased up to 38.6 per cent and with both DC and UC one may reach up to 47.6 per cent[11]. In the following sections, we will discuss UC and DC for a couple of rare-earth ions and examine their application in c-Si based solar cells.

7.1.1 Photon absorption and emission theory Before going into the details of UC and DC processes, it might be useful to recall some well established and relevant interactions and processes, such as ground state absorption (GSA), exited state absorption (ESA) and spontaneous emission (SPE). These processes are sketched in Fig. 7. 1. 56Chapter 7. Up and Down Conversion and Their Application to Solar Cells

FIGURE 7.1: Absorption and emission process between two energy levels.

To determine the output intensity of a UC system, we can consider a system with an number of incoming photons Np that all have an energy ∆Eab. The density of atoms in the ground and the excited state are Na and Nb, and their energies are Ea and Eb, respectively [87]. The energy difference ∆Eab is given by: ∆Eab = Eb − Ea (7.1) The absorption of photons within this system is proportional to the number of incoming photons with energy ∆Eab . Photons that can be absorbed by the ground state are given by: dNp = −Bba Nb Np (7.2) dt abs where Bba is the Einstein coefﬁcient that depends on the properties of states Na and Nb.The negative sign indicates that the photon number decreases as the photons are absorbed. The stimulated emission is given by: dNp = Bba Nb Np (7.3) dt stim 7.2. Up and Down Conversion 57

Finally, the spontaneous emission is not dependent on the incoming photons. It is given by: dNp = ANb (7.4) dt spon A is the Einstein coefﬁcient, which is related to the radiative life time. Under steady state conditions, the total number of photons in the system is given by: dNp dNp dNp + + = 0 (7.5) dt abs dt stim dt spon

The ratio of the population density of the upper level Nb and lower level Na is described by Boltzmann distribution function

N ∆E a = exp (7.6) Nb kbT

Where kb is Boltzmann constant and T is the prevailing temperature. Solving for Np, we obtain: A Np = (7.7) B Na − 1 ba Nb The general total emission is given by: dNp = Bba Nb(Np + 1) (7.8) dt tot This theory helps to calculate the ﬂux caused by converted photons. The extra ﬂux Φ over the length d of the material sample is given by: dNp Φ = d (7.9) dt tot

7.2 Up and Down Conversion

UC converts two or more photons into a higher energetic photon for harvesting the sub-band-gap photons and adding them to the visible spectrum that may be absorbed by a solar cell and generate extra current [11, 12, 88]. Therefore, UC provides an attractive method of reducing the spectral losses and improving overall efﬁciency of the solar cells. This enhancement from an UC material can be applied to all kinds of solar cells.

DC is the reverse process of UC, where higher energy photons are converted to lower energy photons [89]. DC is usually driven by ultraviolet photons. About 35 per cent of the solar energy that reaches the earth’s surface is lost by thermal effect due to the mismatch with the band-gap of the semiconductor material. The DC layer in solar cells will absorb ultraviolet (higher energy) photons and split them into photons closer to the band gap 58Chapter 7. Up and Down Conversion and Their Application to Solar Cells energy. The same theory for calculating the current density from UC, can be applied to DC. The ideal QE of DC should be 200 per cent, because one photon is converted to at least two photons.

The UC/DC materials are to be implemented as extra layers. The UC layer should be placed under the solar cell, because low energetic photons can not be absorbed and are transmitted through the solar cell. After UC the higher energetic photons will be reﬂected at the back contacts and delivered into the active layer of a solar cell. A DC layer should be placed on the top of a solar cell to absorb more high energetic photons and convert them into the visible range before reaching the active layer of the solar cell. Fig. 7. 2 illustrates the structure of solar cell with frequency conversion layers.

FIGURE 7.2: Schematic view of a solar cell with UC and DC layers.

7.3 Rate Equations for Up/Down-Conversion

7.3.1 Up-Conversion Case We are going to examine some rare-earth ions contained in co-doped glass layers. Typical rare earth ions are Erbium Er+3, Neodymium Nd3+, Yb3+ and Thulium Tm3+. These ions are well-known rare-earth elements with favourable energy levels that allow for photo emission within the visible range of the solar spectrum. The term diagram for a typical example of UC is shown by Fig. 7. 3., which highlights the most important optical processes. There are basically three typical transitions, called ground state absorption (GSA) 3 3 3 3 ( H6 → H5 transition), ﬁrst excited state absorption (1 ESA) ( H4 → F6 3 1 transition) and second excited state absorption (2 ESA) ( F4 → G4 transition), which includes a number of metastable states are shown below. 7.3. Rate Equations for Up/Down-Conversion 59

FIGURE 7.3: UC energy level diagram with the most important processes for Tm3+ in the ﬂuoride glass [90].

The rate equations for this schema are given by:

dN 0 = −W N + A N + A N + A N dt 1 0 10 1 30 3 50 5 dN 1 = −(A + W + W )N + A N + W N dt 10 2 s 1 21 2 5 3 dN 2 = W N − A N + A N dt 1 0 21 2 52 5 dN3 (7.10) = W N − (A + W + W )N + A N dt s 1 30 s 3 3 43 1 dN 4 = W N − A N dt 2 1 43 4 dN 5 = W N − (A + A )N dt 3 3 50 52 5 NTm+3 = N0 + N1 + N3 + N5

Here Wi is the probability of the pumping transition i, denotes the initial energy level and j, denotes the ﬁnal level. Aij is the corresponding decay rate. The pumping transition, Wi, is proportional to the pump power ,Pp which is given by: Wi = Ppσi/Ep Ae f f (7.11)

Where σi is the absorption cross-section, Ep is the pump energy and Ae f f is the effective cross-section. The population density of N2 and N4 are usually negligible for such a schema [90]. In the steady state (dNi/dt = 0), the population densities , N0, N1, N3 and N5 can be derived from the equations as a function of the pump power. Provided that all necessary constants in Eq. 7. 10 are known from experimental results. Note that Aij can be calculated by Judd-Ofelt theory as well. The important parameters needed to solve for the population densities are given in table 7. 1 [90]. 60Chapter 7. Up and Down Conversion and Their Application to Solar Cells

Parameter Symbol Value

−21 2 GSA cross section σ1 1.1 × 10 cm

−19 2 1 ESA cross section σ2 8.2 × 10 cm

−20 2 2 ESA cross section σ3 1 × 10 cm

−19 2 stimulation cross section σs 6.7 × 10 cm

−1 Decay rate from the level 1 to 0 A10 108.6 s

−1 Decay rate from the level 2 to 0 A30 594.6 s

−1 Decay rate from the level 5 to 0 A50 491.8 s

−1 Decay rate from the level 5 to 2 A52 384.4 s

26 −3 Total population density NTm+3 5 × 10 m

Pump power Pp 0.2 W

TABLE 7.1: Parameters for solving the rate equations of Eq. 7. 10.

For Tm3+ ions in the ﬂuoride glass, we neglect energy transfer between the ions (i.e. assuming that the ions are far enough from each other) we analysed the corresponding up-conversion scheme. Fig. 7. 3 shows infrared input photons of 1064nm and up-converted output photons of 480 nm, 750 nm and 800 nm at visible wavelengths. The initial intensity used in the simulation model was the AM1.5 spectral intensity averaged in the UC window to ef- fectively 1064 nm in the infrared range. As we can see from Eqs. 7. 10 and 7. 11 the probability of a transition Wi is proportional to the pump power and the population density of the i state. We will solve the rate equation with a given realistic pump power. To this end Fig. 7. 4 shows the population density as a function of this pump power. Finally we choose here the effective cross-section as Ae f f = 0.5 × 0.5 µm for calculate Wi [91]. 7.3. Rate Equations for Up/Down-Conversion 61

FIGURE 7.4: Population density of state N1, N3 and N5 as a function of pump power for the UC system of Eq. 7. 10.

In Fig. 7. 4, the population densities are proportional to the pump power, and a higher power gives a high population density. But the population density of 26 −3 the ground state N0 is almost constant and equal to 4.99 × 10 m .

7.3.2 Down-Conversion Case To illustrate down conversion, we use the system as, Td3+-Yb3+ co-doped in ﬂuoroindogallate glass (DC layer). Note that there is an energy transfer (ET) between the two rare-earth ions. With this type of conversion layer, photons can be converted from the ultraviolet range (360 nm) to the near-infrared range (980 nm). The processes of ground state absorption (GSA) and (ET) are shown in Fig. 7.5. 62Chapter 7. Up and Down Conversion and Their Application to Solar Cells

FIGURE 7.5: DC energy level diagram with the important processes for Tb3+-Yb3+.

The rate equation for this schema are given by:

dN 1 = −W N + A N + CN N dt 13 1 21 2 2 4 dN 2 = −A N − CN N − A N dt 21 2 2 4 32 2 dN 3 = −A N + W N dt 32 3 13 1 dN (7.12) 4 = A N − CN N dt 54 5 2 4 dN 5 = CN N − A N dt 2 4 54 5 NTB+3 = N1 + N2 + N3 NYb+3 = N4 + N5 Again these rate equations are solved under steady state conditions for a given total population density. The parameters for solving the rate equations are summarised in table 7. 2. 7.4. Simulation of an Ideal c-Si Solar Cell With and Without UP/Down 63 Conversion

Paramiter Symbol Value

−26 2 GSA cross section σ13 1.38 × 10 m

−1 Decay rate from the level 2 to 1 A21 150.19 s

−1 Decay rate from the level 3 to 2 A32 48030 s

−1 Decay rate from the level 5 to 4 A54 10 s

ET coefﬁcient C 2 × 10( − 21) m−3

3+ 25 −3 Total population density of Tb NTb3+ 3 × 10 m

3+ 25 −3 Total population density of Yb NTb3+ 3 × 10 m

TABLE 7.2: Parameters for solving the rate equations of DC [92].

Note that C is the ET cross-relaxation coefﬁcient that relates to concentration of the rare-earth ions [13, 93].

Fig.7.5 shows that the input photons have a wavelength of 360 nm and the output photons have a wavelength of 980 nm, corresponding to DC from an ultraviolet wavelength to a near visible wavelength.

With parameters from table 7.1., 7.2., we solve the rate equations we 21 −3 obtained for DC N5 = 8.9 × 10 m . The assumed layer thickness, effective 2 cross-section and pump power are d = 3 mm, Ae f f = 0.25 mm and Pp = 0.2 W for DC.

7.4 Simulation of an Ideal c-Si Solar Cell With and Without UP/Down Conversion

The additional photon ﬂux density from UC or DC, φedd,UC/DC over the length d of the conversion material can be estimated by

φedd,UC/DC = Aij Nid (7.13)

Where Ni is the population density of a state i, from which the electrons decay to a lower electronic state j and emit the converted photons and Aij is decay rate from state i to state j. This additional photon ﬂux must be added to the incoming photon ﬂux from the Sun. It will then generate additional charge carriers in the active region of the solar cell. For a single emission line this 64Chapter 7. Up and Down Conversion and Their Application to Solar Cells additional generation rate is given by

Gadd = αφadd,UC/DC (7.14) where α is the absorption coefﬁcient of the solar cells at that speciﬁc wavelength. The additional current density due to the conversion layer or material can be estimated using Jadd = qGaddLD (7.15) where q is the charge of the electron and LD is the estimated width of the depletion layer. The net current density of the of the augmented solar cell is then given by: Jnet = Jdark − Jp − Jadd (7.16)

Here dark current density, Jdark and photocurrent density Jp were already mentioned and calculated in the previous chapter. The last term of the equation, Jadd is the additional current density coming from the conversion material.

The results from UC and DC and their effect on the solar cell performance are summarized in the table 7. 3.

Parameter Symbol

−2 Jp 42.2 Am

−2 Jadd,UC 5.76 mAcm

−2 Jadd,DC 2.14 mAcm

α at 800 nm and 980 nm ≈ 1 × 105 and ≈ 0.5 × 105 m−1

−6 LD 10 × 10 m

TABLE 7.3: Performance of the c-Si solar cell with the UP/DC conversion layer.

In relation to the results obtained above, it is important to calculate the efﬁ- ciency of a solar cell augmented by an additional conversion material layer. Fig. 7. 6 shows the J-V curves of the ideal c- Si solar cell with and without such an additional conversion material layer. 7.4. Simulation of an Ideal c-Si Solar Cell With and Without UP/Down 65 Conversion

FIGURE 7.6: J-V for ideal c-Si solar cell, and with UC/DC materials.

The percentage of efﬁciency enhancement ∆η was calculated using the following expression V × J − V × J ∆η = mC mC m m · 100% (7.17) Pin −2 Here Pin = 1000 Wm is the input solar power, Vm and Jm are the voltage and current density at the maximum power point of the J − V curve, and VmC and JmC are the voltage and the current density at the maximum power point of the of the solar cell augmented by an UC or DC layer, which can be determined from a J − V curve. In the UC case, the infrared photons at our pump wavelength of 1064 nm are utilized by the UC layer and extra photons are produced at the corresponding (λ = 800 nm) emission wavelength. For DC ultraviolet photons at a pump wavelength 360 nm are utilised by the DC layer to generate photons at the corresponding (980 nm) emission wavelength. We further assume that each absorbed photon generates one electron-hole pair (ηc = 1). Then the efﬁciency enhancement for UC and DC are ∆ηUC ≈ 3% and ∆ηDC ≈ 2.1%.

However note that these efficiencies are strongly dependent on the size of the frequency conversion layers which are fluoride glass for the UC layer and fluoroindogallate glass for DC layer. For our two examples of UC and DC, a different glass layer thickness leads to significant differences in the results (in the percentage range). On the other hand the thicker the glass 66Chapter 7. Up and Down Conversion and Their Application to Solar Cells layers are, the more UC or DC photons they will absorb and less UC or DC photons will reach the active layer of the solar cell. This absorption may in principle be implemented into the present approach using Beer’s law with a suitable absorption coefficient α(λ) but for this thesis, we are looking at a method that provides a certain pre-screening of suitable UC and DC schemes, rather than predicting the results of real experiments.

The present approach is only valid for very thin an very transparent glass layer, but it offers a swift method to check the validity and plausibility of experimental results and to suggest new UC and DC system. The goal must be to ﬁnd a system, which for example offers a 5-10 per cent increase in efﬁciency for glass layers in the µm range or less. 67

Chapter 8

Conclusions and Future Work

8.1 Conclusions

We have sketched the physics of a solar cell device and pointed out the limitations of conventional solar cells based on Si. Various processes have been experimentally shown to improve the efficiency of a solar cell like plasmonic nanoparticles and up/down conversion. This would allow efficiencies higher than the Shockley-Queisser limit for a single junction solar cell [9]. In this thesis, we have developed and applied models to implement these new features in a solar cell device simulation, using a simple diode model for c-Si as a starting point. Next, we have focused on strategies to harness light within the infrared and ultraviolet frequency ranges based on UC/DC (up- conversion/down-conversion) processes [8]. We used rare-earth ions and co-doped glass such as Erbium Er+3 , Neodymium Nd3+ , Yb3+ and Thulium Tm3+ as an example for UC/DC. Parameters such as the transition lifetimes for the UC and DC processes can be calculated from Judd-Ofelt theory or taken from experimental results. We used rate equations to determine the population density relating to the most promising optical processes in an UC or DC layer. The corresponding photon flux can be used to calculate the extra photo-currents arising from these frequency conversion processes. We showed how to add these photo currents from a single emission line to a generic current model of a typical P-N junction, and estimated the resulting increases in efficiency. The efficiency enhancement of UC and DC was calculated to about 3 per cent and 2.1 per cent respectively but for relatively thick glass layers.

We also noted in our simulation that the rates for the generation of free charges by the incoming sun light are strongly determined by the absorption properties of the basic semiconductor materials used for the p-n junction. These corresponding absorption coefficients may either be taken from experimental or first principles calculations. We also indicated that virtually the whole set of materials parameters necessary to perform device simulations can either be taken from experiment, or they can be also be determined by first principles simulations. We also showed examples of the corresponding device simulations.

We also simulated the absorption and scattering of light using nanoparticles 68 Chapter 8. Conclusions and Future Work between 1 − 100 nm diameter which show strong dipolar excitation known as surface plasmon resonances. In particular for thin film solar cells, plasmonic nanoparticles such as Au and Ag can be used to increase the absorption of thin film solar cells. These nanoparticles also scatter light into the active layer of a solar cell, and thus increase the overall efficiency of such as device. We simulated and discussed several scenarios with an up to 5 per cent increase in efficiency, but also presented a model for the heating of a solar cell due to light absorption effects of these nanoparticles, which would reduce the overall efficiency.

This thesis must be seen as part of a worldwide effort to push the efficiency of commercial single-junction solar cells into efficiency ranges, which have only been achieved by expensive complex multi-junction solar cells up to now [85]. Many groups in the world are competing in this field, and new progress is made almost every day. We hope that this thesis will be helpful in this regard, and fill the gap between experimental studies of high efficient solar cells, and typical device simulation of conventional solar cells.

8.2 Future Work

In the future, the methods that were used in this thesis can be applied to both new and established materials such as CdTe and Perovskites. These materials have been studied by many groups worldwide[94]. All the parameters that are needed for the simulation of the corresponding solar cells can be calculated by ab-initio methods. Other rare-earth ions and host materials can also be explored in the future, which might be used to develop more efficient frequency conversion schemes in the far infrared or extreme UV range. The Up and Down conversion method can identify new and multiple rare-earth ions to drastically improve solar cell device performance. The methods used for calculating current density related to UC/DC schemes may be related to Beer’s law to get more accurate results. We can try different methods to convert photons to energies close the band gap energies to increase the efficiencies of solar cells for example quantum dots. Note that the method that we used for UC/DC was the effect of only one single wavelength photon result. But the solar spectrum has a large range of wavelengths, in the future we may be able to integrate over the UV and IR range of spectrum which can highly increase efficiency in solar cells. 69

Appendix A

Python Codes for Plotting

A.1 Dielectric Function Based on the Drude- Lorentz Model

1 #Imaginary and Real part of the dielectric function 2 #for silver and gold using the Drude−Lorentz model 3 #see Eq. 4.2 and Fig. 4.3 4 import numpy as np 5 import pylab as pl 6 from numpy. lib .scimath import logn 7 from math importe 8 9 s o l = 3e8# speed of ligthm/s 10 t p i = 2 * np . pi 11 l = np.linspace(250e −9, 1600e −9, 200)# wavelength(m) 12 13 #dielectric function for gold 14 def epsilon_Au(l): 15 hb = 6.582119514e−16# hbar 16 w = t p i * s o l / l# omega 17 epsi = 6.8890#infinite dielectric constant 18 f1 = 1.7857#Lorentz resonant term 19 wp = 8.9601 / hb# plasma frequency 20 w1 = 2.9715 / hb# Lorentz resonance frequency 21 gamma = 0.0723 / hb#damping term proportional 22 gamma1 = 0.9503/hb#Lorentz oscillator damping rate 23 return epsi−wp**2 / (w**2+1 j *gamma*w)−f1 *w1** 2 / (w**2−w1**2+1 j * gamma1*w) 24 25 #dielectric function for silver 26 def epsilon_Ag(l): 27 hb = 6.582119514e−16# hbar 28 w = t p i * s o l / l# omega 29 wp = 9.2093 / hb# plasma frequency 30 w1 = 4.2840 / hb#Lorentz resonance frequency 31 gamma = 0.020 / hb#damping term proportional 32 gamma1 = 0.3430/hb#Lorentz oscillator damping rate 33 f1 = 0.4242#Lorentz resonant term 34 epsi = 3.7180#infinite dielectric constant 35 return epsi−wp**2 / (w**2+1 j *gamma*w)−f1 *w1** 2 / (w**2−w1**2+ 1 j * gamma1*w) 36 37 epsr = np.real(epsilon_Au(l)) 38 epsi = np.imag(epsilon_Au(l)) 70 Appendix A. Python Codes for Plotting

39 epsgr = np.real(epsilon_Ag(l)) 40 epsgi = np.imag(epsilon_Ag(l)) 41 f, (pl2, pl1) = pl.subplots(1, 2) 42 pl1.plot(l, epsr, color="black", linewidth=3, linestyle=" −−", label ="Au") 43 pl1.plot(l, epsgr, color="black", linewidth=3, linestyle=" −", label ="Ag") 44 pl2.plot(l, epsi, color="red", linewidth=3, linestyle=" −−", label=" Au") 45 pl2.plot(l, epsgi, color="red", linewidth=3, linestyle=" −", label=" Ag") 46 pl1.legend(loc=’down right’) 47 pl2.legend(loc=’upper right’) 48 49 pl1.ticklabel_format(style=’sci’, axis=’x’, scilimits=(1,0)) 50 pl2.ticklabel_format(style=’sci’, axis=’x’, scilimits=(1,0)) 51 pl1.set_xlabel(r’$Wavelength(m)$’, size=15 ) 52 pl2.set_xlabel(r’$Wavelength(m)$’, size=15 ) 53 pl2.set_ylabel(r’$\epsilon_{Drude −Lorentz}$’, size=20) 54 pl1.set_title(r’Re($\epsilon$)’, fontsize=20) 55 pl2.set_title(r’Im($\epsilon$)’, fontsize=20) 56 pl . show ( ) LISTING A.1: Python example A.2 Scattering and Absorption by Metal Nanopar- ticles

1 #Scattering and absorption cross −s e c t i o n for Au and 2 #Ag Nps asa function of the wavelength, and for various radius 3 #of the metallic nanoparticles. see Eqs. 4. 3, 4.4 and Fig. 4.5 4 5 import numpy as np 6 import pylab as pl 7 from numpy. lib .scimath import logn 8 from math importe 9 10 s o l = 3e8# speed of ligthm/s 11 t p i = 2 * np . pi 12 13 l1 = np.linspace(400e −9, 900e −9, 300)# wavelength(m) 14 l2 = np.linspace(262e −9, 600e −9, 300) 15 16 #dielectric function for gold 17 def epsilon_Au(l): 18 hb = 6.582119514e−16# hbar 19 w = t p i * s o l / l# omega 20 epsi = 6.8890#infinite dielectric constant 21 f1 = 1.7857#Lorentz resonant term 22 wp = 8.9601 / hb# plasma frequency 23 w1 = 2.9715 / hb# Lorentz resonance frequency 24 gamma = 0.0723 / hb#damping term proportional 25 gamma1 = 0.9503/hb#Lorentz oscillator damping rate 26 return epsi−wp** 2 / (w**2+1 j *gamma*w)−f1 *w1**2 / (w**2−w1**2+1 j * gamma1*w) 27 A.2. Scattering and Absorption by Metal Nanoparticles 71

28 #dielectric function for silver 29 def epsilon_Ag(l): 30 hb = 6.582119514e−16# hbar 31 w = t p i * s o l / l# omega 32 wp = 9.2093 / hb# plasma frequency 33 w1 = 4.2840 / hb#Lorentz resonance frequency 34 gamma = 0.020 / hb#damping term proportional 35 gamma1 = 0.3430/hb#Lorentz oscillator damping rate 36 f1 = 0.4242#Lorentz resonant term 37 epsi = 3.7180#infinite dielectric constant 38 return epsi−wp**2 / (w**2+1 j *gamma*w)−f1 *w1** 2 / (w**2−w1**2+ 1 j * gamma1*w) 39 40 # polarizability 41 def alpha_Au(l): 42 eps0 = 1 43 epsm = 1 . 8#dielectric constant of surrounding medium 44 Rnp = 80e−9#radius of the nanoparticles 45 eps = epsilon_Au(l) 46 return4 *np . pi * eps0 *Rnp ** 3 * ( eps − epsm)/(epsm + eps) 47 # polarizability 48 def alpha_Ag(l): 49 eps0 = 1 50 epsm = 1 . 8#dielectric constant of surrounding medium 51 Rnp = 40e−9#radius of the nanoparticles 52 eps = epsilon_Ag(l) 53 return4 *np . pi * eps0 *Rnp ** 3 * ( eps − epsm)/(epsm + eps) 54 55 #m ** 2 , Absorption cross−s e c t i o n Au 56 def abs_Au(l1): 57 k = t p i / l 1#wavevector in the medium 58 o1 = alpha_Au(l1) 59 eps0 = 1 60 returnk * np.imag(o1)/eps0 61 62 #m ** 2 , scattering cross−s e c t i o n Au 63 def sct_Au(l1): 64 k = t p i / l 1#wavevector in the medium 65 o1 = alpha_Au(l1) 66 eps0 = 1 67 returnk ** 4 * ( np .abs(o1)) ** 2 / ( 6 * np . pi * eps0 ** 2 ) 68 69 #m ** 2 , Absorption cross−s e c t i o n Ag 70 def abs_Ag(l2): 71 k = t p i / l 2#wavevector in the medium 72 o1 = alpha_Ag(l2) 73 eps0 = 1 74 returnk * np.imag(o1)/eps0 75 76 #m ** 2 , scattering cross−s e c t i o n Ag 77 def sct_Ag(l1): 78 k = t p i / l 1#wavevector in the medium 79 o1 = alpha_Ag(l1) 80 eps0 = 1 81 returnk ** 4 * ( np .abs(o1)) ** 2 / ( 6 * np . pi * eps0 ** 2 ) 82 83 aau = abs_Au(l1) 72 Appendix A. Python Codes for Plotting

84 sau = sct_Au(l1) 85 aag = abs_Ag(l2) 86 sag = sct_Ag(l2) 87 f, (pl2, pl1) = pl.subplots(1, 2) 88 pl1.plot(l1, aau, color="red", linewidth=3, linestyle=" −", label=" abs") 89 pl1.plot(l1, sau, color="black", linewidth=3, linestyle=" −−", label ="scatt") 90 pl2.plot(l2, aag, color="red", linewidth=3, linestyle=" −", label=" abs") 91 pl2.plot(l2, sag, color="black", linewidth=3, linestyle=" −−", label ="scatt") 92 pl1.legend(loc=’down right’) 93 pl2.legend(loc=’upper right’) 94 pl2.ticklabel_format(style=’sci’, axis=’x’, scilimits=(1,0)) 95 pl2.ticklabel_format(style=’sci’, axis=’y’, scilimits=(1,0)) 96 pl1.ticklabel_format(style=’sci’, axis=’x’, scilimits=(1,0)) 97 pl1.ticklabel_format(style=’sci’, axis=’y’, scilimits=(1,0)) 98 pl1.set_xlabel(r’$Wavelength(m)$’, size=15 ) 99 pl2.set_ylabel(r’$\sigma(m^2)$’, size=20) 100 pl2.set_xlabel(r’$Wavelength(m)$’, size=15 ) 101 pl1.set_title(r’Au $r=80nm$’, fontsize=20) 102 pl2.set_title(r’Ag $r=40nm$’, fontsize=20) 103 pl . show ( ) LISTING A.2: Python example A.3 Heat Generated by Metal Nanoparticles

1 #Temperature map based Au nanoparticles 2 #see Eq. 4.12 and Fig. 4. 11 3 from mpl_toolkits.mplot3d import axes3d 4 from matplotlib import cm 5 from matplotlib.ticker import LinearLocator , FormatStrFormatter 6 import matplotlib.pyplot as plt 7 import numpy as np 8 import matplotlib as mpl 9 s o l = 3e8#m/s, light speed 10 k0 = 0 . 6#w/mK, k0 is thermal conductivity 11 I0 = 1e8#w/m ** 2 , I0=c * E_0 ** 2 * eps0 ** 0 . 5/8 * pi, light i n t e n s i t y 12 hb = 6.582119514e−16# hbar 13 w = 2*np . pi * s o l / 520e−9# omega 14 epsi = 6.8890#infinite dielectric constant 15 f1 = 1.7857#Lorentz resonant term 16 wp = 8.9601 / hb# plasma frequency 17 w1 = 2.9715 / hb# Lorentz resonance frequency 18 gamma = 0.0723 / hb#damping term proportional 19 gamma1 = 0.9503/hb#Lorentz oscillator damping rate 20 21 ndim = 50 22 # total number of NPs in complex 23 N = 12 24 #K, initial temperature 25 T0 = 271 26 eps0 =1.8 27 E0=np. sqrt (8 * np . pi * I0 /( s o l *np.sqrt(eps0))) A.3. Heat Generated by Metal Nanoparticles 73

28 #(spherical gold particle radius)m 29 Rnp = np.array([20e −9,20e −9,20e −9,20e −9,20e −9,20e −9,20e −9,20e −9,20e −9,20e −9,20e −9, 20e −9]) 30 V_Np = 4*np . pi *Rnp**2/3 31 epsilon_Au = epsi−wp** 2 / (w**2+1 j *gamma*w)−f1 *w1**2 / (w**2−w1**2+1 j * gamma1*w) 32 #Q is absorbed power 33 Q = w* ( np .abs(3 * eps0 /(2* eps0+epsilon_Au) ) ) ** 2 * np.imag(epsilon_Au) * E0 ** 2 / ( 8 * np . pi ) 34 #temperature distribution 35 dT = V_Np*Q / (4 * np . pi * k0 ) 36 37 X = np.linspace(−300e −9, 300e −9, ndim ) 38 Y = np.linspace(−300e −9, 300e −9, ndim ) 39 40 X, Y = np.meshgrid(X, Y) 41 42 X0 = np.array([[50e −9] ,[150 e −9] ,[50 e −9] ,[150 e −9] ,[ −50e −9] ,[ −150e −9] ,[ −50e −9] ,[ −150e −9] ,[ −50e −9] ,[ −150e −9] ,[50 e −9] ,[150 e −9]]) 43 Y0 = np.array([[0],[0],[100e −9] ,[100 e − 9],[0],[0],[100e −9] ,[100 e −9] ,[ −100e −9] ,[ −100e −9] ,[ −100e −9] ,[ −100e −9]]) 44 45 T = np.zeros((ndim, ndim)) 46 #T += T0 47 48 f o ri in range(N): 49 T += dT[ i ] * Rnp[i] / np.sqrt(np.abs(((X − X0 [ i ] ) ** 2 . 0 + (Y − Y0 [ i ] ) ** 2 . 0 ) ) ) 50 51 52 fig = plt.figure() 53 ax = fig.add_subplot(111, projection=’3d’) 54 ax = plt.gca() 55 56 57 surf = ax.plot_surface(X, Y, T, cmap=plt.cm.coolwarm, rstride=1, cstride=1,linewidth=0, antialiased=False) 58 ax.view_init(45, −45) 59 60 Tmax = np . max(T) 61 62 mpl.rcParams.update({’font.size’: 16}) 63 ax.zaxis. set_major_locator(LinearLocator(1)) 64 ax. zaxis . set_major_formatter(FormatStrFormatter(’%.0f’)) 65 66 fig.colorbar(surf , shrink = 0.5, aspect=10) 67 ax.set_xlabel(r’$X(m)$’ ,size=20) 68 ax.set_ylabel(r’$Y(m)$’ ,size=20) 69 ax.set_zlabel(r’$\DeltaT(K)$’ ,size=20) 70 p l t . show ( ) 71 p l t . c l f LISTING A.3: Python example

Appendix B

Publications And Conferences/Workshops Attended

B.1 Publications

? T. Aslan, I. Mokgosi, R. Warmbier and A. Quandt, Solar cell device simulations, Transparent Optical Networks (ICTON). IEEE Xplore, Spain, 2017.

? T. Aslan, I. Mokgosi, R. Warmbier and A. Quandt, Advanced light harnessing features in solar cell device simulations, Transparent Optical Networks (ICTON). IEEE Xplore, Spain, 2018. (Submitted)

? T. Aslan, I. Mokgosi, R. Warmbier, A. Quandt, M. Ferrari and G.Righini, About the implementation of up -conversion processes in solar cell simulations, Glassy Materials Based Microdevices. MDPI, Switzerland, 2018. (submitted) 76 Appendix B. Publications And Conferences/Workshops Attended

B.2 Conferences/workshops Attended

? PLESC international workshop, Plasmonics and nanoantennas for solar cells, held on 09/05/2017 at University of the Witwatersrand, Johannesburg/South Africa.

? MERG annual workshop, held on 01/11/2017-02/11/2017 at Soci- eta Dante Alighieri, Johannesburg/South Africa.

? Programme MERG workshop and SA-Italy bilateral project, held on 9.9.2015 at Sunwa River Lodge, Parys/Free State/South Africa. 77

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